International Journal of Group Theory
Volume:9 Issue: 1, Mar 2020

For a finite group group‎, ‎denote by mathcal V(G) the smallest positive integer k with the property that the probability of generating G by k randomly chosen elements is at least 1/e. Let G be a finite soluble group‎. ‎{Assume} that for every pin pi(G) there exists G_pleq G such that p does not divide |G:G_p| and {mathcal V}(G_p)leq d. Then {mathcal V}(G)leq d+7.‎

‎‎A subgroup H of a group G is called malonormal in G if H cap H^x =langle 1rangle for every element x notin N_G(H)‎. ‎These subgroups are generalizations of malnormal subgroups‎. ‎Every malnormal subgroup is malonormal‎, ‎and every selfnormalizing malonormal subgroup is malnormal‎. ‎Furthermore‎, ‎every normal subgroup is malonormal‎. ‎In this paper we obtain a description of finite and certain infinite groups‎, ‎whose subgroups are malonormal‎.

‎Generalizing the concept of quasinormality‎, ‎a subgroup H of a group G is said to be 4-quasinormal in G if‎, ‎for all cyclic subgroups K of G‎, ‎langle H,Krangle=HKHK‎. ‎An intermediate concept would be 3-quasinormality‎, ‎but in finite p-groups‎ - ‎our main concern‎ - ‎this is equivalent to quasinormality‎. ‎Quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups‎, ‎particularly in finite‎ ‎p-groups‎. ‎However‎, ‎even in the smallest case‎, ‎when H is a 4-quasinormal subgroup of order p in a finite p-group G‎, ‎precisely how H is embedded in G‎ ‎is not immediately obvious‎. ‎Here we consider one of these questions regarding the commutator subgroup [H,G]‎.

In this survey we present some significant bounds for the‎ ‎number of maximal subgroups of a given index of a finite group‎. ‎As a‎ ‎consequence‎, ‎new bounds for the number of random‎ ‎generators needed to generate a finite $d$-generated group with high‎ ‎probability which are significantly tighter than the ones obtained in‎ ‎the paper of Jaikin-Zapirain and Pyber (Random generation of finite‎ ‎and profinite groups and group enumeration‎, ‎emph{Ann. Math.}‎, ‎textbf{183} (2011) 769--814) are obtained‎. ‎The results of‎ ‎Jaikin-Zapirain and Pyber‎, ‎as well as other results of Lubotzky‎, ‎Detomi‎, ‎and Lucchini‎, ‎appear as particular cases of our theorems‎.

The purpose of this paper is to present a comprehensive overview of known and new results concerning the structure of groups in which all subgroups‎, ‎except those having a given property‎, ‎are either self-centralizing or self-normalizing‎.

‎We summarize several results about non-simplicity‎, ‎solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes‎. ‎We also collect some problems that have only been partially solved‎.