Journal of the Australian Mathematical Society (5)
Bulletin of the Australian Mathematical Society (1)
Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)
Australian Mathematical Society Inc (6)
London Mathematical Society Lecture Note Series (2)
Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras
Representation theory of groups
Structure and classification of infinite or finite groups
Special aspects of infinite or finite groups
General nonassociative rings
S. P. Glasby, Frederico A. M. Ribeiro, Csaba Schneider
Journal: Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View
Published online by Cambridge University Press: 25 February 2019, pp. 1-26
Let p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative 𝔽p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, 𝔽).
ON THE SECOND-LARGEST SYLOW SUBGROUP OF A FINITE SIMPLE GROUP OF LIE TYPE
S. P. GLASBY, ALICE C. NIEMEYER, TOMASZ POPIEL
Journal: Bulletin of the Australian Mathematical Society / Volume 99 / Issue 2 / April 2019
Let $T$ be a finite simple group of Lie type in characteristic $p$ , and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$ -subgroup except for an explicit list of exceptions and that $S$ is always 'large' in the sense that $|T|^{1/3}<|S|\leq |T|^{1/2}$ . One might anticipate that, moreover, the Sylow $r$ -subgroups of $T$ with $r\neq p$ are usually significantly smaller than $S$ . We verify this hypothesis by proving that, for every $T$ and every prime divisor $r$ of $|T|$ with $r\neq p$ , the order of the Sylow $r$ -subgroup of $T$ is at most $|T|^{2\lfloor \log _{r}(4(\ell +1)r)\rfloor /\ell }=|T|^{O(\log _{r}(\ell )/\ell )}$ , where $\ell$ is the Lie rank of $T$ .
EDITORIAL TO THIS SPECIAL ISSUE IN MEMORY OF LACI KOVÁCS
S. P. GLASBY, DANE FLANNERY, CHERYL E. PRAEGER, RALPH STÖHR
Journal: Journal of the Australian Mathematical Society / Volume 102 / Issue 1 / February 2017
Published online by Cambridge University Press: 17 January 2017, pp. 1-2
PRIMITIVE PRIME DIVISORS AND THE $\mathbf{n}$ TH CYCLOTOMIC POLYNOMIAL
Linear algebraic groups and related topics
Finite fields and commutative rings (number-theoretic aspects)
Computational number theory
S. P. GLASBY, FRANK LÜBECK, ALICE C. NIEMEYER, CHERYL E. PRAEGER
Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ , which is closely related to the cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{n}(x)$ and to primitive prime divisors of $q^{n}-1$ . Our definition of $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$ , we provide an algorithm for determining all pairs $(n,q)$ with $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)\leq cn^{k}$ . This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.
5 - The shape of solvable groups with odd order
By S. P. Glasby, Department of Mathematics, Central Washington University
Edited by C. M. Campbell, University of St Andrews, Scotland, M. R. Quick, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
Book: Groups St Andrews 2005
Print publication: 04 January 2007, pp 432-437
On the tensor product of polynomials over a ring
Computational aspects and applications
S. P. Glasby
Journal: Journal of the Australian Mathematical Society / Volume 71 / Issue 3 / December 2001
Given polynomials a and b over an integral domain R, their tensor product (denoted a ⊗ b) is a polynomial over R of degree deg(a) deg(b) whose roots comprise all products αβ, where α is a root of a, and β is a root of b. This paper considers basic properties of ⊗ including how to factor a ⊗ b into irreducibles factors, and the direct sum decomposition of the ⊗-product of fields.
Subgroups of the upper-triangular matrix group with maximal derived length and a minimal number of generators
By S P Glasby, The University of the South Pacific
Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, N. Ruskuc, University of St Andrews, Scotland, G. C. Smith, University of Bath
Book: Groups St Andrews 1997 in Bath
Print publication: 18 February 1999, pp 275-281
On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups
Journal: Journal of the Australian Mathematical Society / Volume 58 / Issue 2 / April 1995
If R is a 2-group of symplectic type with exponent 4, then R is isomorphic to the extraspecial group , or to the central product 4 o 21+2n of a cyclic group of order 4 and an extraspecial group, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful representation of R of degree 2n. The 2-cocycle associated with this projective representation takes values which are powers of −1 if R is isomorphic to and powers of otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.
2-groups with every automorphism central
Journal: Journal of the Australian Mathematical Society / Volume 41 / Issue 2 / October 1986
Print publication: October 1986
An infinite family of 2-groups is produced. These groups have no direct factors and have a non-abelian automorphism group in which all automorphisms are central.