arXiv subjects
Publications and source records attributed to Gustavo A. Fernández-Alcober.
Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all $i\ge 0$: (i) $\exp Ω_-i(G)\le p^i$ for odd p, and $\exp Ω_-i(G)\le 2^{i+1}$ for p = 2; (ii) the index $|G:G^{p^i}|$ coincides with the number of elements of G of order at most $p^i$.
If G is a finitely generated powerful pro-p group satisfying a certain law v=1, and if G can be generated by a normal subset T of finite width which satisfies a positive law, we prove that G is nilpotent. Furthermore, the nilpotency class of G can be bounded in terms of the prime p, the number of generators of G, the law v=1, the width of T, and the degree of the positive law. The main interest of this result is the application to verbal subgroups: if G is a p-adic analytic pro-p group in which all values of a word w satisfy positive law, and if the verbal subgroup w(G) is powerful, then w(G) is nilpotent.
We prove that outer commutator words are uniformly concise, i.e. if an outer commutator word w takes m different values in a group G, then the order of the verbal subgroup w(G) is bounded by a function depending only on m and not on w or G. This is obtained as a consequence of a structure theorem for the subgroup w(G), which is valid if G is soluble, and without assuming that w takes finitely many values in G. More precisely, there is an abelian series of w(G), such that every section of the series can be generated by values of w all of whose powers are also values of w 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.