Simple groups, maximal subgroups, and probabilistic aspects of profinite groups (Q677424)

The paper is concerned with so called asymptotic group theory and probabilistic aspects of profinite groups. Let \(m_n(G)\) be the number of maximal subgroups of a group \(G\) of index at most \(n\). As the first result the authors show that for every \(\varepsilon>0\) there is a constant \(c(\varepsilon)\) such that \(m_n(G)\leq cn^{1.875+\varepsilon}\) for all almost simple finite groups \(G\) and for all positive integers \(n\). As a corollary the authors prove that a \(d\)-generated finite group has at most \(2n^s\) maximal subgroups of index \(n\), where \(s=s(d)=\max(d,3,1.875+\varepsilon+\log c)\) and \(\varepsilon\) and \(c\) are as above. This corollary is used in the study of generation probabilities in profinite groups. Recall that a profinite group \(G\) is compact, and hence has a Haar measure, which one can normalise so that \(G\) has measure 1, and is thus a probability space. The group \(G\) is said to be positively finitely generated if for some \(k\) the measure \(P(G,k)\) of the set of \(k\)-tuples generating \(G\) is positive. It is proved that a profinite group is positively finitely generated if and only if it is of polynomial maximal subgroup growth. As a consequence the authors deduce that the class of finitely generated residually finite groups of polynomial maximal subgroup growth is extension closed. As a final result the authors show that the number of generators of open subgroups of polynomial subgroup growth profinite group growths rather slowly. Namely, for such a \(G\) there exists a constant \(c\) such that \(d(H)\leq c\sqrt{\log[G:H]}\) for all open subgroups \(H\) of \(G\). Most of the results of the paper rely heavily on the classification of finite simple groups.