Volume 17, Issue 6

Julius Whiston and Jan Saxl [J. Algebra 258 (2002), 651–657] showed that the size of an irredundant generating set of the group G = PSL(2, p ) is at most four and computed the size m ( G ) of a maximal set for many primes. We will extend this result to a larger class of primes, with the surprising result that, when p ¬≡ ±1 mod 10, m ( G ) = 3 except for the special case p = 7. In addition, we will determine which orders of elements in irredundant generating sets of PSL(2, p ) with lengths less than or equal to four are possible in most cases. We also give some remarks about the behavior of PSL(2, p ) with respect to the replacement property for groups.

Local finiteness is proved for groups of exponent dividing 72 with no elements of order 6.

If G is a group which admits a manifold model for B G then G is a Poincaré duality group. We study a generalisation of Poincaré duality groups, introduced initially by Davis and Leary in [High-Dimensional Manifold Topology, World Scientific, Singapore (2003), 139–150], motivated by groups G with cocompact manifold models M for E̠ G where M H is a contractible submanifold for all finite subgroups H of G . We give several sources of examples and constructions of these Bredon–Poincaré duality groups, including using the equivariant reflection group trick of Davis and Leary to construct examples of Bredon–Poincaré duality groups arising from actions on manifolds M where the dimensions of the submanifolds M H are specified. We classify Bredon–Poincaré duality groups in low dimensions, and discuss behaviour under group extensions and graphs of groups.

We generalize the classical Post correspondence problem (𝐏𝐂𝐏 n ) and its non-homogeneous variation (𝐍𝐏𝐂𝐏 n ) to non-commutative groups and study the computational complexity of these new problems. We observe that 𝐏𝐂𝐏 n is closely related to the equalizer problem in groups, while 𝐍𝐏𝐂𝐏 n is connected to the double twisted conjugacy problem for endomorphisms. Furthermore, it is shown that one of the strongest forms of the word problem in a group G (we call it the hereditary word problem ) can be reduced to 𝐍𝐏𝐂𝐏 n in G in polynomial time. The main results are that 𝐏𝐂𝐏 n is decidable in a finitely generated nilpotent group in polynomial time, while 𝐍𝐏𝐂𝐏 n is undecidable in any group containing free non-abelian subgroups (though the argument is very different from the classical case of free semigroups). We show that the double endomorphism twisted conjugacy problem is undecidable in free groups of sufficiently large finite rank. We also consider the bounded 𝐏𝐂𝐏 and observe that it is in 𝐍𝐏 for any group with 𝐏-time decidable word problem, meanwhile it is 𝐍𝐏-hard in any group containing free non-abelian subgroups. In particular, the bounded 𝐏𝐂𝐏 is 𝐍𝐏-complete in non-elementary hyperbolic groups and non-abelian right angle Artin groups.

We classify all non-abelian groups G for which there exists a pair ( V , W ) of absolutely simple Yetter–Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional, under two assumptions: the square of the braiding between V and W is not the identity, and G is generated by the support of V and W . As a corollary, we prove that the dimensions of such V and W are at most six. As a tool we use the Weyl groupoid of ( V , W ).

Given natural numbers m and n , we define a deflation map from the characters of the symmetric group S mn to the characters of S n . This map is obtained by first restricting a character of S mn to the wreath product S m ≀ S n , and then taking the sum of the irreducible constituents of the restricted character on which the base group S m × ⋯ × S m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S mn under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan–Nakayama rule and special cases of the Littlewood–Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.

Let p be a prime number, G be a finite group, P be a p -subgroup of G , and k be an algebraically closed field of characteristic p . We prove that the k G -Scott module with vertex P is Brauer indecomposable for some families of groups closely related to groups constructed by Park in the context of fusion systems.

Let G be a nonabelian finite group and let d be an irreducible character degree of G . Then there is a positive integer e so that | G | = d ( d + e ). Snyder has shown that if e > 1, then | G | is bounded by a function of e . This bound has been improved by Isaacs and by Durfee and Jensen. In this paper we will show, for groups having a nontrivial, abelian normal subgroup, that | G | ≤ e 4 - e 3 . Given that there are a number of solvable groups that meet this bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.

In [J. Pure Appl. Algebra 2 (1972), 371–393, Theorem 4.1], J. A. Green shows that the Green Correspondence in Finite Group Modular Representation Theory is a consequence of an equivalence between two quotient categories of appropriate subcategories in the Green Correspondence context. In [Adv. Math. 104 (1994), 297–314, Theorems 3.5, 3.6 and 3.7], M. Auslander and M. Kleiner prove a similar result. M. Linckelmann suggested that the quotient categories in these results are the same. Utilizing extensions of [The Representation Theory of Finite Groups, North-Holland, Amsterdam, 1982, III, Theorem 7.8] or [Representations of Finite Groups, Academic Press, San Diego, 1988, Chapter 5, Corollary 3.11], we extend these results to blocks of finite groups. In order to state and prove our results and to remain relatively self-contained, we follow the procedures of [Adv. Math. 104 (1994), 297–314] in the Green Correspondent context. This is presented in Section 1. In Section 2 we present our main results. In Section 3 we give a very short proof of a theorem of H. Fitting for 𝒪-algebras that is essential in the proof of basic results of J. A. Green, [J. Pure Appl. Algebra 2 (1972), 371–393, Lemma 3.9 and Theorem 3.10].