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Leire Legarreta

Publications and source records attributed to Leire Legarreta.

9 recordsLinked to original sources
arXiv2020

Metabelian groups: full-rank presentations, randomness and Diophantine problems

We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that if $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\geq |A|$. We further prove that if $|R| \leq |A|-1$ then in any direct decomposition of $G$ all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

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↗