Volume 26, Issue 5

In this article, we compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups. We show that the generating function of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups differs from that in degree 0 by a rational function.

We describe the Stiefel–Whitney classes (SWCs) of orthogonal representations 𝜋 of the finite special linear groups G = SL ⁡ ( 2 , F q ) G=\operatorname{SL}(2,\mathbb{F}_{q}) , in terms of character values of 𝜋. From this calculation, we can answer interesting questions about SWCs of 𝜋. For instance, we determine the subalgebra of H * ⁢ ( G , Z / 2 ⁢ Z ) H^{*}(G,\mathbb{Z}/2\mathbb{Z}) generated by the SWCs of orthogonal 𝜋, and we also determine which 𝜋 have non-trivial mod 2 Euler class.

Let 𝐺 be a finite permutation group on Ω. An ordered sequence ( ω 1 , … , ω ℓ ) (\omega_{1},\ldots,\omega_{\ell}) of elements of Ω is an irredundant base for 𝐺 if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of 𝐺 have the same cardinality, 𝐺 is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups 𝐺 to the case that the socle of 𝐺 is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.

Let 𝑁 be a normal subgroup of a finite group 𝐺. Let N ≤ H ≤ G N\leq H\leq G such that 𝑁 has a complement in 𝐻 and ( | N | , | G : H | ) = 1 (\lvert N\rvert,\lvert G:H\rvert)=1 . If 𝑁 is abelian, a theorem of Gaschütz asserts that 𝑁 has a complement in 𝐺 as well. Brandis has asked whether the commutativity of 𝑁 can be replaced by some weaker property. We prove that 𝑁 has a complement in 𝐺 whenever all Sylow subgroups of 𝑁 are abelian. On the other hand, we construct counterexamples if Z ⁢ ( N ) ∩ N ′ ≠ 1 \mathrm{Z}(N)\cap N^{\prime}\neq 1 . For metabelian groups 𝑁, the condition Z ⁢ ( N ) ∩ N ′ = 1 \mathrm{Z}(N)\cap N^{\prime}=1 implies the existence of complements. Finally, if 𝑁 is perfect and centerless, then Gaschütz’ theorem holds for 𝑁 if and only if Inn ⁢ ( N ) \mathrm{Inn}(N) has a complement in Aut ⁢ ( N ) \mathrm{Aut}(N) .

In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .

In this paper, we prove that the Morse boundary of a free product depends only on the Morse boundary of its factors. In fact, we also prove the analogous result for graphs of groups with finite edge groups and infinitely many ends. This generalizes earlier work of Martin and Świątkowski in the case of non-hyperbolic groups.

We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group, we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.

We give a flexible method for constructing a wide variety of limits of betweenness relations. This unifies work of Adeleke, who constructed a Jordan group preserving a limit of betweenness relations, and Bhattacharjee and Macpherson who gave an alternative method using a Fraïssé-type construction. A key ingredient in their work is the notion of a tree of 𝐵-sets. We employ this and extend its use to a wider class of examples.