In this paper, we review the behaviour of the Frattini subgroup Φ(G) and the Frattini factor group G/Φ(G) of an infinite group G having in mind the most standard results of the finite case. Este art́ıculo está dedicado con todo cariño a la memoria de Chicho. Al no formar parte de tu grupo de investigación, no tuve la fortuna de relacionarme contigo por este motivo, pero śı que tuve la oportunida...

We prove that for any prime number p, every finite non-abelian p-group G of class 2 has a noninner automorphism of order p leaving either the Frattini subgroup Φ(G) or Ω1(Z(G)) elementwise fixed.

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, oft...

We prove that the number of groups of order p whose Frattini subgroup is central is for fixed n a PORC (‘polynomial on residue classes’) function of p. This extends a result of G. Higman.

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, oft...