On an analogue to the Grushko-Neumann theorem involving finite groups

Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of profinite groups is whether or not $d(G)=d(H)+d(K),$ which is an analogue to the Grushko-Neumann theorem for abstract free products of groups.Ribes and Wong show that the two following assertions are equivalent: (1) For every pair H and K of pro-C-groups, $d(H\ *\ K)=d(H)+d(K).$ (2) For every pair of H and K of finite groups in C, there is a finite group G in C such that $G={},\ {\rm and}\ d(G)=d(H)+d(K).$Using the classification of finite simple groups, we provide evidence that assertion (2) in the case that C consists of all finite groups is false for certain H and K. This might be thought of as an extension of the evidence provided so far by Kovacs, Sim, and Lucchini.U of I OnlyETDs are only available to UIUC Users without author permissio