MR2169820
Motose, Kaoru(J-HIROST-MS); Tsugita, Hitoshi(J-HIROST-MS)
Permutation polynomials in several variables over finite fields.
Bull. Fac. Sci. Technol. Hirosaki Univ. 8 (2005), no. 1, 7--10.
11T06 (05Exx 12E20)

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{A review for this item is in process.}

MR2134778
Motose, Kaoru(J-HIROST-MS)
Gauss sums are just characters of multiplicative groups of finite fields.
Proceedings of the 37th Symposium on Ring Theory and Representation Theory, 55--59,
Symp. Ring Theory Represent Theory Organ. Comm., Osaka, 2005.
11T24 (11L05)

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR2134769 (2005j:16001) .}


Previous Item

MR2083730 (2005d:11172)
Motose, Kaoru(J-HIROST-MS)
On values of cyclotomic polynomials. VII.
Bull. Fac. Sci. Technol. Hirosaki Univ. 7 (2004), no. 1, 1--8.
11T22 (11R18)

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The author studies the factorization of the $n$-th cyclotomic polynomial $\Phi_n(X)$ over an arbitrary field $K$, where $\Phi_n(X)$ is the monic polynomial whose roots are the primitive roots of unity of highest order dividing $n$ in an algebraic closure of $K$. He also gives a proof of a known result about the roots of unity in a number field. Then, for $\Phi_n$ over the rational numbers, he shows that $\Phi_n(x)$ is a strictly increasing function of the real variable $x$, for $x \geq 2$, and for $x \geq 1$ under certain conditions, thus correcting a statement in one of his previous papers \ref[Part I, Math. J. Okayama Univ. 35 (1993), 35--40 (1995); MR1329911 (96j:11167)]. He then studies a sequence of polynomials in two variables with rational integer coefficients satisfying a recurrence relation which, for special values of the variables, gives the Fibonacci sequence, and he relates them to the cyclotomic polynomials.

\{For Part VI see \ref[K. Motose, Bull. Fac. Sci. Technol. Hirosaki Univ. 6 (2004), no. 2, 1--5; MR2047419 (2004m:11181)].\}

Reviewed by Charles Helou
Previous Item

MR2077013
Motose, Kaoru(J-HIROST-MS)
Let's use cyclotomic polynomials in your lecture for your students.
Proceedings of the 36th Symposium on Ring Theory and Representation Theory, 96--101,
Symp. Ring Theory Represent Theory Organ. Comm., Yamanashi, 2004.
12E20

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR2079172 (2005a:16002) .}


Previous Item

MR2047419 (2004m:11181)
Motose, Kaoru(J-HIROST-MS)
On values of cyclotomic polynomials. VI.
Bull. Fac. Sci. Technol. Hirosaki Univ. 6 (2004), no. 2, 1--5.
11R18 (11R29)

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The author gives proofs of well-known results about the discriminant and the ring of integers of cyclotomic fields. The main tool used consists of some properties of cyclotomic polynomials, established in a previous paper of the author \ref[Part V, Math. J. Okayama Univ. 45 (2003), 29--36 MR2038836 (2004k:11187)].

Reviewed by Charles Helou
Previous Item

MR2038836 (2004k:11187)
Motose, Kaoru(J-HIROST-MS)
On values of cyclotomic polynomials. V.
Math. J. Okayama Univ. 45 (2003), 29--36.
11T22 (11T71 11Y05 12E10 12E20)

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The author uses properties of cyclotomic polynomials to give new proofs of some known results on finite fields and a method of factorization for integers and suggests a cyclic code. The main tool is that, in a commutative ring $R$ of characteristic $l >0$, if an element $\alpha$ satisfies $\phi_n (\alpha) =0$, where $n$ is a positive integer and $\phi_n$ is the $n$-th cyclotomic polynomial, then $n$ is equal to a power of $l$ multiplied by the order of $\alpha$ in $R$. One result on factorization thus deduced is that, for positive integers $a, m, n$, if gcd$(am , n) =1$ and $a^m \equiv 1 \pmod n$, then $n = \prod_{d \mid m} (n, \phi_d(a))$.

\{For Part IV see [K. Motose, Bull. Fac. Sci. Technol. Hirosaki Univ. 1 (1998), no. 1, 1--7; MR1659463 (2000c:11196)].\}

Reviewed by Charles Helou
Previous Item

MR2003540 (2004f:11083)
Motose, Kaoru(J-HIROST-MS)
On Gauss sums and Vandermonde matrices.
Bull. Fac. Sci. Technol. Hirosaki Univ. 6 (2003), no. 1, 19--23.
11L05 (11C20)

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The author provides two proofs of quadratic reciprocity. The proofs are united by their reliance on the discriminant of the cyclotomic polynomial $x^{p-1}+x^{p-2}+\dots+x+1$. Both proofs are short and easy to follow. They make use of several basic elements of algebraic number theory, particularly the properties of Vandermonde determinants, character sums and the Legendre symbol.

Let $A$ be the Vandermonde matrix used in evaluating the discriminant of the cyclotomic polynomial described above. By using properties of the trace and determinant of $A$ the author also proves the following: $$ g(\eta)=\sum_{k=0}^{p-1} \zeta^{k^2}=i^{\frac{(p-1)^2}{4}} p, $$ where $\eta$ is the quadratic character of the multiplicative group of the finite field of characteristic $p$, and $g(\eta)$ is the Gauss sum associated with $\eta$.

Reviewed by Jason Douglas Rosenhouse
Previous Item

MR2001568 (2004i:16062)
Motose, Kaoru(J-HIROST-MS)
On the nilpotency index of the radical of a group algebra. XII.
Bull. Fac. Sci. Technol. Hirosaki Univ. 6 (2003), no. 1, 25--29.
16Y30

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From the text: "We gave negative answers in \ref[J. London Math. Soc. (2) 25 (1982), no. 1, 39--42; MR0645863 (83d:20003)] to Tsushima's problem \ref[Y. Tsushima, in Proceedings of the 10th Symposium on Ring Theory (Shinshu Univ., Matsumoto, 1977), 116--120, Dept. Math., Okayama Univ., Okayama, 1978; MR0472895 (57 \#12581) (Problems 3, 4, 5)]. This example was an affine group over a finite field. In this paper, we show that a finite field, which is the foundation of this affine group, can be extended to a finite Dickson near field." \{For Part XI see [K. Motose, Math. J. Okayama Univ. 44 (2002), 51--56 (2003); MR1961124 (2004d:16049)].\}
Previous Item

MR1969473
Motose, Kaoru(J-HIROST-MS)
On the nilpotency index of the radical of a group algebra.
Proceedings of the 35th Symposium on Ring Theory and Representation Theory (Okayama, 2002), 161--164,
Symp. Ring Theory Represent Theory Organ. Comm., Okayama, 2003.
16S34

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR1969451 (2003k:16001) .}


Previous Item

MR1961124 (2004d:16049)
Motose, Kaoru(J-HIROST-MS)
On the nilpotency index of the radical of a group algebra. XI.
Math. J. Okayama Univ. 44 (2002), 51--56 (2003).
16S34 (16N99)

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This is the eleventh part of a long sequence of papers. Let $K$ be a field of characteristic $p>0$ and let $G$ be a finite $p$-solvable group. If $t(G)$ is the nilpotency index of the radical of the group algebra $K[G]$, then it is known that $t(G)\geq e(p-1)+1$, where $p^e$ is the $p$-part of $|G|$. In this paper, the author uses Dickson near fields to construct an example of a group $G$ of $p$-length 2, with $O_{p',p}(G)/O_p(G)$ nonabelian, and with $t(G)=e(p-1)+1$.

\{Part X has been reviewed \ref[K. Motose, Math. J. Okayama Univ. 34 (1992), 57--66 (1994); MR1272606 (95c:16034)].\}

Reviewed by Donald S. Passman
Previous Item

MR1851207 (2002j:11149)
Motose, K.(J-HIROST-MS)
On values of cyclotomic polynomials. (English. English summary)
International Symposium on Ring Theory (Kyongju, 1999), 231--234,
Trends Math.,
Birkhäuser Boston, Boston, MA, 2001.
11T22 (11A51 11T71)

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Cyclotomic polynomials of order $n \geq 1$ are defined as $$ \Phi_n(x) = \prod_{(k,n) = 1} (x - \zeta_n^k), $$ where $\zeta_n$ is a primitive (complex) $n$th root of unity, and $k$ runs over positive integers less than $n$ that are relatively prime to $n$. These polynomials are known to have integer coefficients; thus for integer values of $x$, these numbers are integers, referred to as cyclotomic numbers.

The article under review is a survey of results obtained by the author in four earlier papers \ref[Math. J. Okayama Univ. 35 (1993), 35--40 (1995); MR1329911 (96j:11167); Math. J. Okayama Univ. 37 (1995), 27--36 (1996); MR1416242 (97h:11151); Math. J. Okayama Univ. 38 (1996), 115--122 (1998); MR1644477 (99g:11144); Bull. Fac. Sci. Technol. Hirosaki Univ. 1 (1998), no. 1, 1--7; MR1659463 (2000c:11196)]. These results include interesting properties of cyclotomic numbers that are applied to give primality tests that extend the Lucas and Pepin tests.

Cyclotomic numbers are relevant for cryptography since, for large values of $n$ and $x$, the prime factorizations of these numbers provide instances of primes with a large number of digits. Further, the result that a divisor $d$ of the cyclotomic number $\Phi_n(a)$, with $(d,n)=1$, is an $a$-pseudo-prime, is used by the author to show that pseudo-primes which occur in this way can be used to construct one-way ciphers, based on the idea of construction of an RSA cipher. Such a cipher (described in this article) avoids the need for two large primes required in the RSA cipher, and is based on knowledge of factors of cyclotomic numbers.

An ongoing project by Morimoto et al. aims to provide complete prime factorization tables for the cyclotomic numbers $\Phi_n(x)$, for $2 \leq x \leq 1000$, and $2 \leq \phi(n) \leq 200$, where $\phi$ is the Euler-totient function [cf. Factorization of Cyclotomic Numbers (Japanese), Sophia Univ., 1987; Zbl 0632.10001; II, 1989; Zbl 0661.10001; III (English, Japanese), 1992; Zbl 0778.11001; IV, 1999].

{For the entire collection see MR1851189 (2002c:16002).}

Reviewed by Narasimhan Anuradha
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MR1831075 (2002f:12014)
Motose, Kaoru(J-HIROST-MS)
On finite Dickson near fields.
Bull. Fac. Sci. Technol. Hirosaki Univ. 3 (2001), no. 2, 69--78.
12K05

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This paper basically repeats the proofs of the well-known classification of finite Dickson nearfields, including the structure of the multiplicative group and the automorphism group. The exposition is not better than what is available.

Indeed, everything is more elegant and clear in H. Wähling's book Theorie der Fastk\"orper [Thales, Essen, 1987; MR0956467 (90e:12024)]. For those who don't read German, there's an English summary in that book.

And there are in fact other books, in English, containing most of this material \ref[e.g., H. Lüneburg, Translation planes, Springer, Berlin, 1980; MR0572791 (83h:51008)].

Reviewed by Hubert Kiechle
Previous Item

MR1690922 (2000c:11008)
Motose, Kaoru(J-HIROST-MS)
On commutative group algebras. III.
Bull. Fac. Sci. Technol. Hirosaki Univ. 1 (1999), no. 2, 93--97.
11A15 (11Y11 16S34)

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The author continues his program of proving some number-theoretic results involving characters, by using commutative group algebras. This is a sequel to two previous papers of the author, with the same titles \ref[Sci. Rep. Hirosaki Univ. 40 (1993), no. 2, 127--131; MR1264414 (95b:11009); Math. J. Okayama Univ. 36 (1994), 23--27 (1995); MR1349019 (96k:11149)]. In the present paper, he gives such proofs for the quadratic, cubic and biquadratic reciprocity laws as well as for a primality test of Lenstra.

Reviewed by Charles Helou
Previous Item

MR1659463 (2000c:11196)
Motose, Kaoru(J-HIROSS-MSS)
On values of cyclotomic polynomials. IV.
Bull. Fac. Sci. Technol. Hirosaki Univ. 1 (1998), no. 1, 1--7.
11T22 (11T71 20G40)

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In two previous papers \ref[Math. J. Okayama Univ. 35 (1993), 35--40 (1995); MR1329911 (96j:11167); Math. J. Okayama Univ. 37 (1995), 27--36 (1996); MR1416242 (97h:11151)], the author established several results about the values of cyclotomic polynomials. In this paper, he applies some of those results to obtain a cryptography system similar to RSA, to establish differently a property used in a proof of Wedderburn's theorem for finite division rings, and to obtain some evaluations concerning the orders of certain linear groups over finite fields.

Reviewed by Charles Helou
Previous Item

MR1644477 (99g:11144)
Motose, Kaoru(J-HIROSS)
On values of cyclotomic polynomials. III.
Math. J. Okayama Univ. 38 (1996), 115--122 (1998).
11T22 (11A51)

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The author gives criteria of primality for rational integers of certain types in terms of values of cyclotomic polynomials and of recurrent sequences of algebraic integers. They extend some tests of Lucas and Pépin. This is a sequel to two previous papers by the author \ref[Part I, Math. J. Okayama Univ. 35 (1993), 35--40 (1995); MR1329911 (96j:11167); Part II, Math. J. Okayama Univ. 37 (1995), 27--36 (1996); MR1416242 (97h:11151)].

Reviewed by Charles Helou
Previous Item

MR1416242 (97h:11151)
Motose, Kaoru(J-HIROSS)
On values of cyclotomic polynomials. II.
Math. J. Okayama Univ. 37 (1995), 27--36 (1996).
11T22 (11A07)

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This is a sequel to a previous paper by the author \ref[Part I, Math. J. Okayama Univ. 35 (1993), 35--40 (1995); MR1329911 (96j:11167)]. The author establishes several divisibility properties of values, for integers, of cyclotomic polynomials $\phi_n(X)$. For example, let $a,b,n,s$ be positive integers, with $a,b,n\geq 2$, and $p$ an odd prime not dividing $n$; then $p^s$ is a common divisor of $\phi_n(a)$ and $\phi_n(b)$ if and only if the order of $a\bmod p$ is $n$ and $a^n\equiv 1\bmod p^s$ and $b\equiv a^k\bmod p^s$ for some positive integer $k$ prime to $n$. He also deduces some known results of Cipolla, Lucas and Pépin concerning Mersenne, Fermat or Sophie Germain primes.

Reviewed by Charles Helou
Previous Item

MR1349019 (96k:11149)
Motose, Kaoru(J-HIROS)
On commutative group algebras. II.
Math. J. Okayama Univ. 36 (1994), 23--27 (1995).
11T24 (16S34)

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The author continues the project begun in Part I \ref[Sci. Rep. Hirosaki Univ. 40 (1993), no. 2, 127--131; MR1264414 (95b:11009)] which aims at proving standard results involving characters and equations by a method using commutative group algebras. In the present paper this is done for the prime ideal decomposition of the values of Gaussian sums over finite fields and for the explicit formula for the Gaussian sum with the quadratic character.

Reviewed by Harald Niederreiter
Previous Item

MR1329911 (96j:11167)
Motose, Kaoru(J-HIROSS)
On values of cyclotomic polynomials.
Math. J. Okayama Univ. 35 (1993), 35--40 (1995).
11T22

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From the text: "It is very important to study values of cyclotomic polynomials, especially values for integers (cyclotomic numbers), because important theorems have been proved using properties of these values, for example, Wedderburn's theorem for finite division rings, Artin's theorem for the orders of simple groups and Bang's theorem, a simple proof of which is given in Section 2 of this paper \ref[see H. N. Shapiro, Introduction to the theory of numbers, Wiley, New York, 1983; MR0693458 (84f:10001) (see p. 221)].

"In the Galois theory of commutative rings, I. Kikumasa and T.\ Nagahara proved that the polynomials $f_n(x)=\sum_{d\mid n}µ(d)x^{n/d}$ are strictly increasing functions for $x\geq1$, where $µ$ is the Möbius function, and they used essentially this property in their paper \ref[Proc. Amer. Math. Soc. 115 (1992), no. 3, 593--600; MR1081697 (92i:12006)]. We show in Section 1 of this paper that cyclotomic polynomials $\Phi_n(x)$ have the same property."
Previous Item

MR1264414 (95b:11009)
Motose, Kaoru(J-HIROSS)
On commutative group algebras.
Sci. Rep. Hirosaki Univ. 40 (1993), no. 2, 127--131.
11A15 (11D79 13A99 51M15)

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The author uses commutative group algebras to give a quick proof of the Frobenius determinant relation, one more proof of the law of quadratic reciprocity, an alternative approach to the characterization of regular polygons constructible by ruler and compass, and a proof of a well-known formula for the number of solutions of diagonal equations over finite fields.

Reviewed by Harald Niederreiter
Previous Item

MR1272606 (95c:16034)
Motose, Kaoru(J-HIROSS)
On the nilpotency index of the radical of a group algebra. X.
Math. J. Okayama Univ. 34 (1992), 57--66 (1994).
16S34 (16N40)

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This is one of a long series of papers on the nilpotency index of the Jacobson radical of the group algebra $K[G]$. Here the author considers certain subgroups of the group of semilinear transformations on a finite field $F$ of the same characteristic as that of $K$. Specifically, it is assumed that $G$ contains the full set of translations $U=\{x\mapsto x+a| a\in F\}\cong F\sp+$. The computations are simplified by using a basis for the radical $JK[U]$ determined by linear characters $\lambda\colon F\sp*\to K\sp*$.

Reviewed by Donald S. Passman
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MR1099842
Motose, Kaoru(J-HIROS)
On the nilpotency index of the radical of a group algebra.
Proceedings of the 23rd Symposium on Ring Theory (Chiba, 1990), 95--98,
Okayama Univ., Okayama, 1990.
16N40

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR1099830 (91m:16003) .}


Previous Item

MR1051302 (91d:20003)
Motose, Kaoru(J-HIROS)
On Loewy series of group algebras of some solvable groups.
J. Algebra 130 (1990), no. 2, 261--272.
20C05 (16S34)

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Let $p$ be a prime and $q=p^r$ a $p$-power. Consider the affine semilinear group $\roman A\Gamma(q^p)=\{x\to ax^\sigma+ b$: $a,b\in\roman{GF}(q^p)$, $a\neq 0$, $\sigma\in\roman{Gal} (\roman{GF}(q^p)/\roman{GF}(q))\}$, and let $K$ be a field containing $\roman{GF}(q)$. For groups of $p$-length 1, the Loewy series of the projective indecomposable modules are fairly well understood \ref[see W. Schwarz, same journal 60 (1979), no. 1, 51--75; MR0549098 (81a:20012)]. Since $\roman A \Gamma(q^p)$ has $p$-length 2, it makes sense to investigate its projective indecomposables in greater detail. This was done by \n Y. Ninomiya\en \ref[Math. J. Okayama Univ. 29 (1987), 11--51, 53--75; MR0936729 (89j:20016)] for $\roman A\Gamma (3^3)$, and an algorithm for $\roman A\Gamma(p^p)$ was given by \n U. Stammbach, R. Staszewski\en and the reviewer \ref[Comm. Algebra 17 (1989), no. 5, 1249--1274; MR0993401 (90c:20019)].

The paper under review handles the case in which $G$ is the normal subgroup of $\roman A\Gamma(q^p)$ of index $q-1$. Set $h=(q^p-1)/(q-1)$, and for $s,t\in\{0,1,\cdots,h-1\}$, define an equivalence relation by $s\sim t$ if and only if $s= tq^k\bmod h$ for some $k\in\{0,1,\cdots,p-1\}$. Then the irreducible modules $M_s$ can be labelled by the equivalence classes with respect to $\sim$. For $x\in\bold N$, write $x^*=\sum_ki_k$, where $x=\sum_ki_kp^k$ is the $p$-adic expansion of $x$. If $S^*=t$, define $E_{ts}=\{x\in\{0,1,\cdots,q^p-1\}$: $x^*=t$ and $x\sim s\}$, and set $a_{ts}=|E_{ts}|$ for $s\neq 0$ and $a _{t0}=p|E_{t0}|$. Then the multiplicity of $M_s$ in the $n$th Loewy layer of the principal indecomposable $G$-module equals $(\sum _k a_{n-k,s})/p$, where $k$ runs through $0\leq k\leq p-1$, $0\leq n-k\leq rp(p-1)$ (Theorem 5.2).

A similar result holds for the other projective indecomposables (Theorem 6.5). We do not formulate the result since more technical notation is needed to do so.

\{For Part II see the following review.\}

Reviewed by Olaf Manz
Previous Item

MR1024749 (91d:20004)
Motose, Kaoru(J-HIROS)
On Loewy series of group algebras of some solvable groups. II.
Sci. Rep. Hirosaki Univ. 36 (1989), no. 1, 1--8.
20C05 (16S34)

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Let $q$ be a $p$-power, $\roman A\Gamma(q^p)$ the group of affine semilinear mappings on $\roman{GF} (q^p)$ and $G$ its normal subgroup of index $q-1$. In Part I \ref[J. Algebra 130 (1990), no. 2, 261--272; see the preceding review], the Loewy series of the projective indecomposable $G$-modules were determined. The second part extends those results to arbitrary groups $H$ with $G\leq H\leq \roman A\Gamma(q^p)$. At the end, several examples are listed that were obtained by computer-aided calculations.

Reviewed by Olaf Manz
Previous Item

MR0996247 (89k:16001)
Proceedings of the 21st Symposium on Ring Theory.
Held at Hirosaki University, Hirosaki, October 20--22, 1988. Edited by Kaoru Motose.
Okayama University, Department of Mathematics, Okayama, 1989. vi+71 pp.
16-06

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\{The Twentieth Symposium has been reviewed [MR0924404 (88i:16002)].\} \

Contents:\ Hiroshi Yoshimura, Endomorphism rings of semi-\break cocritical modules (pp. 1--6); Hisaaki Fujita, A counterexample to Tarsy's conjecture on global dimension of orders (pp. 7--13); Isao Kikumasa and Takasi Nagahara, On primitive elements of Galois extensions of commutative rings (pp. 14--20); Yasuyuki Hirano, Some results on additive groups of rings (pp. 21--28); Yugen Takegahara, The character rings of finite groups and generalized Dedekind sums (pp. 29--38); Yutaka Watanabe, On trace forms of algebras (pp. 39--42); Yumiko Hironaka and Fumihiro Sat\=a, Representations of Hecke algebras and symmetric spaces in case of alternating matrices (pp. 43--51); Tadashi Ikeda, On interior $G$-algebras with the trivial defect group (pp.\ 52--58); Yoshitomo Baba, Note on almost $M$-injectives (pp. 59--63); Shigeto Kawata, Auslander-Reiten quivers and Green correspondence (pp.\ 64--69); Shigeo Koshitani, A remark on modular representation theory (pp. 70--71).

\{The papers will not be reviewed individually.\}
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MR0924411
Motose, Kaoru(J-HIROS)
On Loewy series of group algebras of some solvable groups.
Proceedings of the 20th symposium on ring theory (Okayama, 1987), 61--64,
Okayama Univ., Okayama, 1987.
16A26

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR0924404 (88i:16002) .}


Previous Item

MR0866186 (88a:16020)
Motose, Kaoru(J-HIROS)
Some examples of compressible group algebras and of noncompressible group algebras.
Bull. Austral. Math. Soc. 34 (1986), no. 3, 389--394.
16A26 (20C05)

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Let $Z(R)$ be the centre of a ring $R$. An idempotent $e$ of $R$ is said to be compressible in $R$ if $Z(eRe)=eZ(R)e$. The ring $R$ itself is said to be compressible if every idempotent of $R$ is compressible. The aim of this paper is to give some examples of group algebras which are and which are not compressible. The above examples indicate that the problem of characterizing compressible modular group algebras is highly nontrivial. Among other results, it is shown that if $F$ is a field of characteristic $p$ and $G$ a finite $p$-nilpotent group, then the group algebra $FG$ is compressible.

Reviewed by G. Karpilovsky
Previous Item

MR0860061 (87m:16024)
Motose, Kaoru(J-HIROS)
On the nilpotency index of the radical of a group algebra. IX.
Group and semigroup rings (Johannesburg, 1985), 193--196,
North-Holland Math. Stud., 126,
North-Holland, Amsterdam, 1986.
16A26 (20C05)

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Let $t(G)$ denote the nilpotency index of the radical of $K[G]$ for $G$ a finite group and $K$ a field of characteristic $p>0$. If $P$ is a Sylow $p$-subgroup of $G$, it has apparently been conjectured that $t(P)\geq t(G)$. This paper confirms the conjecture in the case of a particular group of $p$-length 2. The computations are interesting.

\edref {Parts VII and VIII have been reviewed \ref[MR 84j:20013; MR 84j:20014].}

{For the entire collection see MR0860046 (87h:20002).}

Reviewed by Donald S. Passman
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MR0847162 (87j:20013)
Motose, Kaoru(J-OKAY)
On the nilpotency index of the radical of a group algebra. VII.
J. Algebra 101 (1986), no. 2, 299--302.
20C05 (16A26)

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Let $K$ be a field of characteristic $p\ge 3$ and let $G$ be a finite $p$-solvable group with Sylow $p$-subgroup $P$ of order $p^m$. Let $t(G)$ denote the nilpotence degree of the Jacobson radical $J(K[G])$; it is shown here that $t(G)=p^{m-1}$ if and only if $p=3$ and $P\cong M(3)$. This extends a result of \n K. Morita\en \ref[Sci. Rep. Tokyo Bunrika Daigaku Sect. A 4 (1951), 177--194; MR0049909 (14,246a)].

\{Part VI has been reviewed [MR0792959 (86m:20007)]. For part VIII see the following review.\}

Reviewed by Donald S. Passman
Previous Item

MR0792959 (86m:20007)
Motose, Kaoru(J-OKAY)
On the nilpotency index of the radical of a group algebra. VI.
J. Algebra 94 (1985), no. 2, 347--351.
20C05

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Let $K$ be a field of characteristic $p>0$, let $G$ be a finite $p$-solvable group and let $P$ be a Sylow $p$-subgroup of $G$ with $|P|=p^m$. In addition set $M=O_{p^\prime}(G)$, $H=O_{p^\prime, p}(G)$ and let $F/M$ be the Frattini subgroup of $H/M$. The author studies $t(G)$, the nilpotency index of the radical $J(KG)$ of the group algebra $KG$. Specifically he offers an alternate proof of the following result of \n S. Koshitani\en \ref[same journal 94 (1985), no. 1, 106--112 ; MR0789542 (86f:20005)]. Theorem: Assume that $t(G)>p^{m-1}$ and that $G$ is not of $p$-length 1. Then $p=2$, $G/F=\text{Sym}_4$ and $F/M$ is cyclic.

\{Part V has been reviewed [MR0757095 (86c:20009)].\}

Reviewed by Donald S. Passman
Previous Item

MR0779768 (86c:20010)
Motose, Kaoru(J-OKAY)
On a theorem of Y. Tsushima.
Math. J. Okayama Univ. 26 (1984), 11--12.
20C05 (16A26 20D20)

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Let $p$ be a prime and let $G$ be a finite $p$-solvable group with a $p$-Sylow subgroup $P$ of order $p^a (a\geq1)$. Let $t(G)$ be the nilpotency index of the radical of the group algebra of $G$ over a field of characteristic $p$. It is proved that if $P$ is nonabelian and regular and if $t(G)=a(p-1)+1$, then $p$ is a Fermat prime and a 2-Sylow subgroup of $G/O_{p'}(g)$ is nonabelian. This paper is virtually identical to a slightly later paper of the author \ref[Proc. Amer. Math. Soc. 92 (1984), no. 3, 327--328].

Reviewed by D. A. R. Wallace
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MR0759645 (87j:20014)
Motose, Kaoru(J-OKAY)
On the nilpotency index of the radical of a group algebra. VIII.
Proc. Amer. Math. Soc. 92 (1984), no. 3, 327--328.
20C05 (16A26)

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Let $G$ be a finite $p$-solvable group with Sylow $p$-subgroup $P$ of order $p^r$ and let $t(G)$ be the nilpotency degree of the radical of a characteristic $p$ group algebra of $G$. The following lemma is proved: Assume that $G$ has $p$-length at least 2 and that $P$ is regular. If $t(G)<(r+2)(p-1)+1$, then $p$ is a Fermat prime and a Sylow 2-subgroup of $G/O_p(G)$ is nonabelian.

As a consequence, certain groups with $t(G)<(r+2)(p-1)+1$ are characterized.

\{For part VII see the preceding review.\}

Reviewed by Donald S. Passman
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MR0757095 (86c:20009)
Motose, Kaoru(J-OKAY)
On the nilpotency index of the radical of a group algebra. V.
J. Algebra 90 (1984), no. 1, 251--258.
20C05

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In this paper, the author continues his explicit computations of the nilpotency index of the Jacobson radical of a group algebra. Here he considers 3-solvable groups $G$ with Sylow 3-subgroup which is the nonabelian group of order 27 and period 3. If $K$ is a field of characteristic 3, then he shows that the nilpotency index of $J(KG)$ is precisely 9. This turns out to be the crucial case needed to extend a result of \n S. Koshitani\en \ref[same journal 80 (1983), no. 1, 134--144; MR 84f:20009] to the prime 3.

\edref{Parts III and IV have been reviewed \ref[MR 83d:20003; MR 85c:20005].}

Reviewed by Donald S. Passman
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MR0738595 (85k:20025)
Motose, Kaoru(J-OKAY)
On a result of S. Koshitani.
Proc. Edinburgh Math. Soc. (2) 27 (1984), no. 1, 57.
20C05 (16A26)

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From the text: "Let $G$ be a $p$-solvable group with a $p$-Sylow subgroup $P$ of order $p\sp a$ and let $t(G)$ be the nilpotency index of the radical of a group algebra of $G$ over a field of characteristic $p$. The purpose of this paper is to give an elementary proof of the following result of S. Koshitani [same journal (2) 25 (1982), no. 1, 31--34; MR0648896 (84f:16015)]. Proposition: Assume that $p$ is odd and $P$ is metacyclic. If $t(G)=a(p-1)+1$ then $P$ is elementary abelian."
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MR0703459 (84g:16007)
Ferrero, Miguel; Kishimoto, Kazuo(J-SHINS); Motose, Kaoru(J-OKAY)
On radicals of skew polynomial rings of derivation type.
J. London Math. Soc. (2) 28 (1983), no. 1, 8--16.
16A21 (16A05)

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Let $K$ be a ring and $D$ be a derivation of $K$. The authors consider two families of ideals $N_K(\alpha)$ and $\scr D(\alpha)$ of $K$ which are defined inductively for ordinals $\alpha$. The definitions are: $N_K(0)=0$, $\scr D(0)=0$; if $N_K(\alpha)$ and $\scr D(\alpha)$ are defined for all $\alpha<\beta$ then if $\beta=\gamma+1$, $N_K(\beta)$ is the sum of all ideals $i$ such that $i^s\subseteq N_K(\gamma)$ and $\scr D(B)$ is the sum of all ideals $j$ such that $D(j)\subseteq j$ and $j^s\subseteq\scr D(\gamma)$ for some integer $s$; if $\beta$ is a limit ordinal, then $N_K(\beta)=\sum_{\gamma<\beta}N_K(\gamma)$, $\scr D(\beta)=\sum_{\gamma<\beta}\scr D(\gamma)$.

They investigate connections between these families for both the ring $K$ and the ring $R=K[X;D]$ (the skew polynomial ring with multiplication given by $ax=xa+D(a)$, $a\in K$). Among the results they obtain are $N_R(\alpha)\cap K=\scr D(\alpha)$ and $N_R(\alpha)=\scr D(\alpha)[X;D]$. Also, they show that $J(R)=(J(R)\cap K)[X;D]$, where $J$ is the Jacobson radical.

Reviewed by L. G. Makar-Limanov
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MR0701962 (85c:20005)
Motose, Kaoru(J-OKAY)
On the nilpotency index of the radical of a group algebra. IV.
Math. J. Okayama Univ. 25 (1983), no. 1, 35--42.
20C05 (16A26)

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Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ of order $p\sp a$ and let $t(G)$ be the nilpotency index of the Jacobson radical of the group algebra $KG$ with $K$ a field of characteristic $p$. It is known that $t(G) \leq p\sp a$ if either $G$ is $p$-solvable or $P$ is cyclic. Furthermore, if $G$ is $p$-solvable then $t(G)\geq a(p-1)+1$. The author computes $t(G)$ for $G={\rm SL}(2,q)$ with $p$ an odd divisor of $q-1$, and studies $p$-solvable groups with $t(G)=a(p-1)+1$ and with $P$ not elementary abelian.

\{Part III has been reviewed [MR0645863 (83d:20003)].\}

Reviewed by Donald S. Passman
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MR0645863 (83d:20003)
Motose, Kaoru
On the nilpotency index of the radical of a group algebra. III.
J. London Math. Soc. (2) 25 (1982), no. 1, 39--42.
20C05 (16A26)

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Let $G$ be a finite $p$-solvable group with Sylow $p$-subgroup $P$ of order $p^a$. If $K$ is a field of characteristic $p$, let $t(G)$ denote the nilpotency index of the Jacobson radical of $K[G]$. Then it is known that $t(G)\geq a(p-1)+1$ [D. A. R. Wallace, Proc. Edinburgh Math. Soc. (2) 16 (1968/69), 127--134; MR0245700 (39 \#7006)]. The author offers two interesting examples related to this: first, a group $G$ with $P$ a nonregular $p$-group but with $t(G)=a(p-1)+1$; second, a group $G$ with central $p'$-element $s$ such that $t(G)>t(G/\langle s\rangle)$. Examples of this nature were previously known only for $p=2$.

\{Part II has been reviewed [MR0595794 (82c:20011)].\}

Reviewed by Donald S. Passman
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MR0603636 (82i:20007)
Motose, Kaoru
On problems of Ninomiya and of Tsushima.
Proceedings of the 13th Symposium on Ring Theory (Okayama Univ., Okayama, 1980), pp. 80--82,
Okayama Univ., Okayama, 1981.
20C05 (16A26)

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From the text: "Let $G$ be a $p$-solvable group with a $p$-Sylow subgroup $P$ of order $p^a$, $K$ an algebraically closed field of characteristic $p$, $KG$ the group algebra of $G$ over $K$, and $t(G)$ the nilpotency index of the radical $J(KG)$ of $KG$.

"D. A. R. Wallace [Proc. Edinburgh Math. Soc. (2) 16 (1968/69), 127--134; MR0245700 (39 \#7006)] proved the inequality $t(G)\geq a(p-1)+1$. In connection with this inequality, Y. Ninomiya and the author [Hokkaido Math. J. 4 (1975), no. 2, 261--264; MR0372007 (51 \#8224)] presented the next problem: If $t(G)=a(p-1)+1$, then is $P$ elementary? Recently, Y. Tsushima [ Proceedings of the 10th Symposium on Ring Theory\/ (Matsumoto, 1977), pp. 116--120; Okayama Univ., Okayama, 1978; MR0472895 (57 \#12581)] presented the following problem: If $E$ is a $p'$-subgroup contained in the center of $G$, then is $t(G)$ equal to $t(G/E)$?

"In the case $p=2$, these problems were answered in the negative [see the author, Math. J. Okayama Univ. 22 (1980), no. 2, 141--143; MR0595794 (82c:20011)]. In this paper, we give negative answers to these problems for every $p$."

{For the entire collection see MR0603629 (81m:16004).}
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MR0595794 (82c:20011)
Motose, Kaoru
On the nilpotency index of the radical of a group algebra. II.
Math. J. Okayama Univ. 22 (1980), no. 2, 141--143.
20C05 (16A21 16A26)

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Let $K$ be an algebraically closed field of characteristic $p>0$ and let $t(G)$ denote the nilpotency index of the radical of the group algebra $K[G]$. In this paper, a number of interesting examples are constructed in characteristic $p=2$. They start with the symmetric group $S_4$ and its Sylow 2-subgroup $D_8$ which are known to satisfy $t(D_8)=5>4=t(S_4)$. Now let $Q$ be the permutation module over $\text{GF}(3)$ for the action of $S_4$ on the cosets of $D_8$ and define $G=Q\times S_4$. Then $G$ is solvable of order $2^3·3^4$ and it is shown here that $t(G)=5$ while $t(G/Q)=4$. This occurs because $K[Q]$ has a centrally primitive idempotent whose stabilizer in $S_4$ is $D_8$ and $t(G)$ is really determined by properties of these stabilizers. This method also yields a twisted group ring $K^t[H]$ such that the nilpotence degree of $JK^t[H]$ is larger than $t(H)$. These examples are really quite surprising. In particular, they answer in the negative Problems $3'$, 4 and 5 of Y. Tsushima [Proceedings of the 10th Symposium on Ring Theory (Shinshu Univ., Matsumoto, 1977), pp. 116--120, Dept. Math., Okayama Univ., Okayama, 1978; MR0472895 (57 \#12581)].

\{Part I has been reviewed [MR0372007 (51 \#8224)].\}

Reviewed by Donald S. Passman
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MR0573665 (81i:16017)
Asano, Shigemoto; Motose, Kaoru
On QF-$2$ algebras with commutative radicals.
Math. J. Okayama Univ. 22 (1980), no. 1, 17--20.
16A36

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Group algebras (of finite groups over an algebraically closed field) with commutative radicals have been studied by D. A. R. Wallace, S. Koshitani, and others. In particular, Wallace gave a result which determines the structure of blocks of such algebras [see Proc. Glasgow Math. Assoc. 7 (1965), 1--8; MR0178074 (31 \#2332)]. That part of his result which is of interest here may be stated in the following form: Let $A$ be a block of an algebra of the above-mentioned type. If the radical $N$ of $A$ is such that $N^2\neq 0$, then $A$ is a commutative completely primary algebra.

The present paper is devoted to the following generalization of this result to QF-2 algebras, in the sense of R. M. Thrall (see Thrall's paper for definitions [Trans. Amer. Math. Soc. 64 (1948), 173--183; MR0026048 (10,98c)]). Theorem: Let $A$ be a QF-2 algebra over a field $K$, and let $A$ be itself a block. Assume that the radical $N$ of $A$ is commutative, and $N^2\neq 0$. Then $A$ is a completely primary almost symmetric algebra over $K$, and $A/N$ is a (commutative) field. Moreover, if the base field $K$ is perfect, then $A$ is a commutative completely primary symmetric algebra over $K$.

An example is given showing that if $K$ is not perfect, then $A$ may not be commutative. The paper closes with a corollary which classifies weakly symmetric algebras which are themselves blocks. Corollary: Let $A$ be a weakly symmetric algebra over a field $K$ and let $A$ be itself a block. Assume the radical $N$ of $A$ is commutative. Then $A$ is one of the following three types: (1) $A$ is a simple algebra over $K$; (2) $A$ is a full matrix ring over a completely primary weakly symmetric algebra $B$ over $K$ such that the square of the radical of $B$ vanishes; (3) $A$ is a completely primary almost symmetric algebra over $K$ such that $A/N$ is a field.

Reviewed by Denis Floyd
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MR0568696 (81g:16021)
Asano, Shigemoto; Motose, Kaoru
On ${\rm QF}-2$ algebras with commutative radicals.
Proceedings of the 12th Symposium on Ring Theory (Hokkaido Univ., Sapporo, 1979), pp. 58--60,
Okayama Univ., Okayama, 1980.
16A36

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This note announces the following two theorems. (1) Let $A$ be a QF-2 algebra over a field $K$ and let $A$ be itself a block. Assume that the radical, $N$, of $A$ is commutative and $N^2\neq 0$. Then $A$ is a completely primary almost symmetric algebra over $K$ with $A/N$ a commutative field. Moreover, if $K$ is perfect $A$ is commutative and symmetric. (2) If $A$ is a finite-dimensional algebra over an algebraically closed field which is a block, and if the radical is generated, as an ideal, by the radical of the center of $A$, then $A$ is a full matrix algebra over the center of $A$. No proofs are given.

{For the entire collection see MR0568687 (81a:16002).}

Reviewed by S. S. Page
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MR0571797 (81g:16028)
Motose, Kaoru
Notes on the radical of a finite-dimensional algebra.
J. Fac. Sci. Shinshu Univ. 14 (1979), no. 2, 95--97.
16A46 (16A21)

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This note contains two new results. Theorem 1: Let $A$ be a (two-sided) indecomposable finite-dimensional algebra over a field $K$. Suppose that $N=\Pi A$ where $N$ is the radical of $A$ and $\Pi$ is the radical of the center $\Lambda$ of $A$. Then $A$ is primary. Furthermore if $K$ is a splitting field for $A$ then $A$ is a full matrix ring over $\Lambda$. Theorem 3: Let $G$ be a finite group, $G'$ its commutator subgroup and $K$ an algebraically closed field of characteristic $p$. Let $A=KG$. If $\Pi$ (the radical of the center) is an ideal of $KG$ then $G'$ is either a $p$-nilpotent group or a $p$-group.

Reviewed by N. Divinsky
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MR0491796 (58 #10995)
Motose, Kaoru
On a theorem of S. Koshitani.
Math. J. Okayama Univ. 20 (1978), no. 1, 59--65.
16A26

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The author gives an alternate proof of a result of S. Koshitani [Tsukuba J. Math. 1 (1977), 137--148; MR0480617 (58 \#773)] concerning the index of nilpotency of the radical of the group algebra $KG$, where $K$ is a field of characteristic $p$ and $G$ is a $p$-group of order $p^r$. The method of proof utilizes Jennings $M$-series. In particular, he shows that the index of nilpotency takes the secondarily highest value $p^{r-1}+p^{-1}$ if and only if $G$ contains an element of order $p^{r-1}$. Furthermore, it is shown that if the index of nilpotency is $p^{r-1}$ then $G$ is $M(3)$ or an elementary abelian group of order 8. Additional results characterizing the index of nilpotency for various classes of groups are given.

Reviewed by Ralph W. Wilkerson
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MR0444762 (56 #3110)
Motose, Kaoru
On radicals of principal blocks.
Hokkaido Math. J. 6 (1977), no. 2, 255--259.
20C20

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Let $G$ be a finite group and $p$ a fixed prime. Let $B_1$ be the principal block of $G$, $J(B_1)$ the radical of $B_1$, and $u_1$ the degree of the principal indecomposable character corresponding to the 1-character of $G$. It is well known that $\dim J(B_1)\leq(\dim B_1)(1-1/u_1)$. One result of this paper is that the equality holds if and only if $u_1$ is the largest positive eigenvalue of the Cartan matrix of $B_1$. The main result is the following: If the Sylow $p$-subgroups of $G$ are cyclic, then the equality holds if and only if $G$ is $p$-solvable of $p$-length 1. The result is an immediate consequence of E. C. Dade's work on blocks with cyclic defect groups [Ann. of Math. (2) 84 (1966), 20--48; MR0200355 (34 \#251)].

Reviewed by P. Fong
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MR0457485 (56 #15690)
Motose, Kaoru; Ninomiya, Yasushi
On generalized uniserial blocks.
Math. J. Okayama Univ. 19 (1976/77), no. 2, 141--145.
16A26 (20C20)

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Let $K$ be a field of characteristic $p>0$, and $G$ a finite group whose order is divisible by $p$. M. Osima [Proc. Phys. Math. Soc. Japan (3) 24 (1942), 1--9; MR0015113 (7,373a)] stated that the group algebra $KG$ is uniserial if and only if $G$ is $p$-nilpotent and a Sylow $p$-subgroup of $G$ is cyclic. But the proof presented was wrong and was later corrected in Theorem 8 of the article by K. Morita [Sci. Rep. Tokyo Bunrika Daigaku Sect. A4 (1951), no. 88, 177--194]. Let $R$ be an Artinian algebra over $K$. By making use of Theorem 1 of Osima [op. cit.] and Theorem 8 of Morita [op. cit.], the authors formulate an analogue for $RG$. Next, in case $K$ is a splitting field, they consider $KG$. If a defect group $D$ of a block $B$ of $KG$ is cyclic, then it can be shown that the nilpotency index $t(B)$ of the radical of $B$ is not greater than the order $|D|$ of $D$. In the main theorem, the authors prove that $t(B)=|D|$ if and only if $B$ is a generalized uniserial ring.

Reviewed by H. Tominaga
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MR0376837 (51 #13012)
Motose, Kaoru; Ninomiya, Yasushi
On the subgroups $H$ of a group $G$ such that $J(KH)KG\supset J(KG)$.
Math. J. Okayama Univ. 17 (1975), no. 2, 171--176.
20C05

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Let $K$ be a field of characteristic $p>0$, and let $G$ be a finite group. We define three families of subgroups of $G$ as follows: (1) $\scr P(G)$ is the set of all subgroups of $p'$-index; (2) $\scr K(G)$ is the set of all subgroups $H$ with $JK[G]\subseteq JK[H]·K[G]$; (3) $\scr L(G)$ is the set of all subgroups $H$ such that, if $V$ is an irreducible $K[H]$-module, then $V^G$ is a completely reducible $K[G]$-module. The authors show that $\scr L(G)=\scr K(G)\subseteq\scr P(G)$ and, furthermore, the latter inclusion is equality if and only if $\scr K(G)$ contains a Sylow $p$-subgroup of $G$. This completes, in a rather simple manner, earlier work of D. C. Khatri [Trans. Amer. Math. Soc. 186 (1973), 51--63 (1974); MR0327880 (48 \#6222)].

Reviewed by Donald S. Passman
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MR0372007 (51 #8224)
Motose, Kaoru; Ninomiya, Yasushi
On the nilpotency index of the radical of a group algebra.
Hokkaido Math. J. 4 (1975), no. 2, 261--264.
20C05

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The authors study the nilpotency index of the radical of a group algebra of a finite group in certain extreme cases.

Reviewed by Donald S. Passman
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MR0399235 (53 #3086)
Motose, Kaoru
On a theorem of Wallace and Tsushima.
Proc. Japan Acad. 50 (1974), 572--575.
20C05

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Let $G$ be a finite group and $K$ be a field. Let $J(G)$ denote the Jacobson radical of the group ring $KG$. The author proves some inequalities between the $K$-dimension of $J(G)$ and that of $J(N)$, where $N$ is a normal subgroup of $G$. Also, he gave an alternative proof of the theorem of D. A. R. Wallace [Proc. Amer. Math. Soc. 12 (1961), 133--137; MR0121408 (22 \#12146)], assuming $G$ is $p$-solvable.

Reviewed by Y. Tsushima
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MR0379644 (52 #549)
Motose, Kaoru
On C. Loncour's results.
Proc. Japan Acad. 50 (1974), 570--571.
20C05

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The author studies the Jacobson radical of the group of a direct product of two finite groups. He uses the well-known fact that $JK[G\times H]=JK[G]·K[H]+K[G]·JK[H]$.

Reviewed by Donald S. Passman
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