Volume 23, Issue 4

Given a group action on a finite-dimensional CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} cube complex, we give a simple criterion phrased purely in terms of cube stabilisers that ensures that the group satisfies the strong Tits alternative, provided that each vertex stabiliser satisfies the strong Tits alternative. We use it to prove that all Artin groups of type FC satisfy the strong Tits alternative.

Let X be a finite type A n 2 {A_{n}^{2}} -polyhedron, n ≥ 2 {n\geq 2} . In this paper, we study the quotient group ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) {\mathcal{E}(X)/\mathcal{E}_{*}(X)} , where ℰ ⁢ ( X ) {\mathcal{E}(X)} is the group of self-homotopy equivalences of X and ℰ * ⁢ ( X ) {\mathcal{E}_{*}(X)} the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X . We show that not every group can be realised as ℰ ⁢ ( X ) {\mathcal{E}(X)} or ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) {\mathcal{E}(X)/\mathcal{E}_{*}(X)} for X an A n 2 {A_{n}^{2}} -polyhedron, n ≥ 3 {n\geq 3} , and specific results are obtained for n = 2 {n=2} .

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

Let G be a group, x , y ∈ G {x,y\in G} , and H is a normal closure in G of the subgroup generated by all commutators of x , y {x,y} that include more than two occurrences of y . Our main result is a parametrization of the uncollected part of Hall’s collection formula for ( x ⁢ y ) n {(xy)^{n}} and two different collection formulas modulo H in an explicit form. In particular, we found explicit expressions for the exponents of the commutators in Hall’s collection formula for both known cases and some unknown cases. Also we give combinatorial identities, which simplify those expressions and help to investigate their divisibility.

Let w be a group-word. For a group G , let G w {G_{w}} denote the set of all w -values in G and w ⁢ ( G ) {w(G)} the verbal subgroup of G corresponding to w . The word w is semiconcise if the subgroup [ w ⁢ ( G ) , G ] {[w(G),G]} is finite whenever G w {G_{w}} is finite. The group G is an FC ⁢ ( w ) {\mathrm{FC}(w)} -group if the set of conjugates x G w {x^{G_{w}}} is finite for all x ∈ G {x\in G} . We prove that if w is a semiconcise word and G is an FC ⁢ ( w ) {\mathrm{FC}(w)} -group, then the subgroup [ w ⁢ ( G ) , G ] {[w(G),G]} is FC {\mathrm{FC}} -embedded in G , that is, the intersection C G ⁢ ( x ) ∩ [ w ⁢ ( G ) , G ] {C_{G}(x)\cap[w(G),G]} has finite index in [ w ⁢ ( G ) , G ] {[w(G),G]} for all x ∈ G {x\in G} . A similar result holds for BFC ⁢ ( w ) {\mathrm{BFC}(w)} -groups, that are groups in which the sets x G w {x^{G_{w}}} are boundedly finite. We also show that this is no longer true if w is not semiconcise.

In this paper, we continue the study of powerfully nilpotent groups. These are powerful p -groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order p n {p^{n}} . We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to p 6 {p^{6}} .

We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut ⁢ ( G ) {\mathrm{Aut}(G)} on G is bounded from above by a constant. Our main results are the following: Firstly, a finite group G only admits Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits of length at most 3 if and only if G is cyclic of one of the orders 1, 2, 3, 4 or 6, or G is the Klein four group or the symmetric group of degree 3. Secondly, there are infinitely many finite (2-)groups G such that the maximum length of an Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbit on G is 8. Thirdly, the order of a d -generated finite group G such that G only admits Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits of length at most c is explicitly bounded from above in terms of c and d . Fourthly, a finite group G such that all Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits on G are of length at most 23 is solvable.

Let G be a finite group, let b be a G -invariant block with defect group Q of the normal subgroup H of G , and let b ′ ∈ Z ⁢ ( 𝒪 ⁢ N H ⁢ ( Q ) ) {b^{\prime}\in Z(\mathcal{O}N_{H}(Q))} be the Brauer correspondent of b . We show that the bijection between the blocks of G covering b and the blocks of N G ⁢ ( Q ) {N_{G}(Q)} covering b ′ {b^{\prime}} , induced by a G / H {G/H} -graded basic Morita equivalence between the block extensions b ⁢ 𝒪 ⁢ G {b\mathcal{O}G} and b ′ ⁢ 𝒪 ⁢ N G ⁢ ( Q ) {b^{\prime}\mathcal{O}N_{G}(Q)} , coincides with the Harris–Knörr correspondence.

For a saturated fusion system ℱ {\mathcal{F}} on a p -group S , we study the Burnside ring of the fusion system B ⁢ ( ℱ ) {B(\mathcal{F})} , as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring B ⁢ ( S ) {B(S)} . We give criteria for an element of B ⁢ ( S ) {B(S)} to be in B ⁢ ( ℱ ) {B(\mathcal{F})} determined by the ℱ {\mathcal{F}} -automorphism groups of essential subgroups of S . When ℱ {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p -group, we show that the restriction of B ⁢ ( G ) {B(G)} to B ⁢ ( S ) {B(S)} has image equal to B ⁢ ( ℱ ) {B(\mathcal{F})} . We also show that, for p = 2 {p=2} , we can gain information about the fusion system by studying the unit group B ⁢ ( ℱ ) × {B(\mathcal{F})^{\times}} . When S is abelian, we completely determine this unit group.

We explore conditions for the Frobenius–Wielandt morphism to commute with the operations of deflation and tensor induction on the Burnside ring. In doing this, we review the commutativity with induction. The techniques used for induction and tensor induction no longer work for deflation, so, in this case, we make use of tools coming from the theory of biset functors.