Gustavo A. Fernández-Alcober

Outer commutator words are uniformly concise

We prove that outer commutator words are uniformly concise, that is, if an outer commutator word u takes m different values in a group G, then the order of the verbal subgroup u(G) is bounded by a function depending only on m, and not on u or G. This is obtained as a consequence of a structure theorem for the subgroup u(G), which is valid if G is soluble, and without assuming that u takes finitely many values in G. More precisely, there is an abelian series of u(G), such that every section of the series can be generated by values of u all of whose powers are also values of u in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction.

GGS-groups: Order of congruence quotients and Hausdorff dimension

If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients

On p-Groups of Maximal Class, II

G_n=G/\Stab_G(n)

A focal subgroup theorem for outer commutator words

for every n. If G is defined by the vector

On the number of conjugacy class sizes and character degrees in finite -groups

e=(e_1,...,e_{p-1})\in\F_p^{p-1}

Cyclotomy over products of finite fields and combinatorial applications

, the determination of the order of

On p -groups of maximal class. III

G_n

A note on conciseness of Engel words

is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.

Conjugacy classes of non-normal subgroups in finite nilpotent groups

In this paper, we continue the study of the p-groups of maximal class from the point of view of the number of conjugacy classes, which was begun in k16] by A. Vera-Lopez and B. Larrea. A p-group G of order pm is said to be of maximal class if its nilpotency class is m 1. Unless otherwise stated, it is supposed throughout this paper that m b 4. The notation used here is practically the same as in [16]. Furthermore, we introduce the vector

Omega subgroups of powerful p-groups

Let