Volume 14, Issue 4

Let p be a prime, and let Γ denote a Sylow pro- p subgroup of the group of automorphisms of the p -adic rooted tree. By using probabilistic methods, Abért and Virág [J. Amer. Math. Soc. 18: 157–192, 2005] have shown that the Hausdorff dimension of a finitely generated closed subgroup of Γ can be any number in the interval [0, 1]. In the case p = 2, Siegenthaler has provided examples of subgroups of Γ of transcendental Hausdorff dimension which arise as closures of spinal groups. In this paper, we show that the situation is completely different for p > 2, since the Hausdorff dimension of the closure of a spinal group is always a rational number in that case. (Here, spinal groups are constructed over spines with one vertex at every level, as in the case p = 2.) Furthermore, we determine the set S of all rational numbers that appear as Hausdorff dimensions of spinal groups. A key ingredient in our approach to this problem is provided by a general procedure for decomposing spinal groups as a semidirect product, which allows us to reduce to the case of 2-generator spinal groups.

An involution in a finite n -dimensional classical group G over a field of odd order q is called ( α, β )-balanced if the dimension of its fixed point subspace is between αn and βn . Balanced involutions play an important role in recent constructive recognition algorithms for finite classical groups in odd characteristic. For a given sequence of conjugacy classes of balanced involutions in G , a c -tuple ( g 1 , . . . , g c ) is a class-random sequence from 𝒳 if, for each i = 1, . . . , c, g i is a uniformly distributed random element of , and the g i are mutually independent. We show that there is a number c = c ( α, β ) such that for large enough n , for a given such sequence 𝒳 of length c , a class-random sequence from 𝒳 generates a subgroup containing the generalized Fitting subgroup of G with probability at least 1 – q – n .

Groups with commuting inner mappings are of nilpotency class at most 2, but there exist loops with commuting inner mappings and of nilpotency class higher than 2, called loops of Csörgő type . In order to obtain small loops of Csörgő type, we expand our programme from [Drápal and Vojtěchovský, European J. Combin. 29: 1662–1681, 2008] and analyze the following set-up in groups: Let G be a group, Z ⩽ Z ( G ), and suppose that δ : G / Z × G / Z → Z satisfies δ ( x, x ) = 1, δ ( x, y ) = δ ( y, x ) –1 , z yx δ ([ z, y ], x ) = z xy δ ([ z, x ], y ) for all x, y, z ∈ G , and δ ( xy, z ) = δ ( x, z ) δ ( y, z ) whenever { x, y, z } ∩ G ′ is not empty. Then there is μ : G / Z × G / Z → Z with δ ( x, y ) = μ ( x, y ) μ ( y, x ) –1 such that the multiplication x ∗ y = xyμ ( x, y ) defines a loop with commuting inner mappings, and this loop is of Csörgoő type (of nilpotency class 3) if and only if g ( x, y, z ) = δ ([ x, y ], z ) δ ([ y, z ], x ) δ ([ z, x ], y ) is nontrivial. Moreover, G has nilpotency class at most 3, and if g is nontrivial then |G| ⩾ 128, |G| is even, and g induces a trilinear alternating form. We describe all nontrivial set-ups ( G, Z, δ ) with |G| = 128. This allows us to construct for the first time a loop of Csörgő type with an inner mapping group that is not elementary abelian.

We introduce and investigate notions of persistent homology for p -groups and for coclass trees of p -groups. Using computer techniques we show that persistent homology provides fairly strong homological invariants for p -groups of order at most 81. The strength of these invariants, together with some of their elementary theoretical properties, suggest that persistent homology may be a useful tool in the study of prime-power groups. In particular, we ask whether the known periodic structure on coclass trees is reflected in a periodic structure on the persistent homology of p -groups in the trees.

The representation dimension rdim( G ) of a finite group G is the smallest positive integer m for which there exists an embedding of G in GL m (ℂ). In this paper we find the largest value of rdim( G ), as G ranges over all groups of order p n , for a fixed prime p and a fixed exponent n ⩾ 1.