arXiv subjects

Gustavo A. Fernandez-Alcober

Publications and source records attributed to Gustavo A. Fernandez-Alcober.

8 recordsLinked to original sources

arXiv2024

Conciseness on normal subgroups and new concise words from outer commutator words

Let $w=w(x_1,\ldots,x_r)$ be an outer commutator word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables. This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers. Our approach is via the study of values of $w$ on normal subgroups, and in this setting we obtain the following result: if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e. $w(N_1,\ldots,N_r)$, is finite.

math.GR↗

arXiv2016

On self-similar finite $p$-groups

In this paper, we address the following question: when is a finite $p$-group $G$ self-similar, i.e. when can $G$ be faithfully represented as a self-similar group of automorphisms of the $p$-adic tree? We show that, if $G$ is a self-similar finite $p$-group of rank $r$, then its order is bounded by a function of $p$ and $r$. This applies in particular to finite $p$-groups of a given coclass. In the particular case of groups of maximal class, that is, of coclass $1$, we can fully answer the question above: a $p$-group of maximal class $G$ is self-similar if and only if it contains an elementary abelian maximal subgroup over which $G$ splits. Furthermore, in that case the order of $G$ is at most $p^p+1$, and this bound is sharp.

math.GR↗

arXiv2016

Some finiteness conditions on normalizers or centralizers in groups

We consider the following two finiteness conditions on normalizers and centralizers in a group G: (i) |N_G(H):H| is finite for every non-normal subgroup H of G, and (ii) |C_G(x): | is finite for every non-normal cyclic subgroup of G. We show that (i) and (ii) are equivalent in the classes of locally finite groups and locally nilpotent groups. In both cases, the groups satisfying these conditions are a special kind of cyclic extensions of Dedekind groups. We also study a variation of (i) and (ii), where the requirement of finiteness is replaced with a bound. In this setting, we extend our analysis to the classes of periodic locally graded groups and non-periodic groups. While the two conditions are still equivalent in the former case, in the latter the condition about normalizers is stronger than that about centralizers.

math.GR↗

arXiv2015

A finiteness condition on centralizers in locally nilpotent groups

We give a detailed description of infinite locally nilpotent groups G such that the index |C_G (x) : | is finite, for every non-normal cyclic subgroup of G. We are also able to extend our analysis to all non-periodic groups satisfying a variation of our condition, where the requirement of finiteness is replaced with a bound.

math.GR↗

arXiv2015

A finiteness condition on centralizers in locally finite groups

We consider a finiteness condition on centralizers in a group G, namely that |C_G (x) : | is finite for every non-normal cyclic subgroup of G. For periodic groups, this is the same as |C_G (x)| is finite for every non-normal cyclic subgroup of G. We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: |C_G (x) : | < n for every non-normal cyclic subgroup of G, for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups.

math.GR↗

arXiv2013

Some restrictions on normalizers or centralizers in finite p-groups

We study three restrictions on normalizers or centralizers in finite p-groups, namely: (i) |N_G(H) : H| <= p^k for every H non-normal in G, (ii) |N_G( ) : | <= p^k for every non-normal in G, and (iii) |C_G(g) : | <= p^k for every non-normal in G. We prove that (i) and (ii) are equivalent, and that the order of a non-Dedekind finite p-group satisfying any of these three conditions is bounded for p>2. More precisely, we get the best possible bound for the order of G in all three cases, which is |G| <= p^{2k+2}. The order of the group cannot be bounded for p=2, but we are able to identify two infinite families of 2-groups out of which |G| <= 2^{f(k)} for some function f(k) depending only on k.

math.GR↗

arXiv2013

A restriction on centralizers in finite groups

For a given m>=1, we consider the finite non-abelian groups G for which |C_G(g): |<=m for every g in G\Z(G). We show that the order of G can be bounded in terms of m and the largest prime divisor of the order of G. Our approach relies on dealing first with the case where G is a non-abelian finite p-group. In that situation, if we take m=p^k to be a power of p, we show that |G|<=p^{2k+2} with the only exception of Q_8. This bound is best possible, and implies that the order of G can be bounded by a function of m alone in the case of nilpotent groups.

math.GR↗

arXiv2007

Comparison of the Discrete and Continuous Cohomology Groups of a Pro-$p$ Group

We address the following question. For which finitely generated pro-$p$ groups the comparison map $ϕ^2:H_{cont}^{2}(P,\F_p) \to H_{disc}{2}(P,\F_p)$ is an isomorphism? We prove that if $P$ is not finitely presented then $ϕ^2$ is not surjective. Furthermore, if $P$ is finitely presented $ϕ^2$ is an isomorphism if and only if the comparison map $ϕ_2:H^{disc}_{2}(P, \F_p) \to H^{cont}_{2}(P, \F_p)$ of second homology groups is an isomorphism. This is the content of Theorem A. The second main result of the paper is Theorem B, which gives an explicit construction of a cochain from the kernel of $ϕ^2$.

math.GR↗