Residual properties of free products of finite groups. (Q1409752)

If \(S\) is a set of groups, then a group \(G\) is residually \(S\) if the intersection of the kernels of the homomorphisms of \(G\) onto members of \(S\) is trivial. The authors' main theorem here is the following. Let \(A\) and \(B\) be non-trivial finite groups, not both 2-groups. Then there is an integer \(r\) depending only on \(A\) and \(B\) such that if \(S\) is any infinite collection of finite simple classical groups of rank at least \(r\), then the free product \(A*B\) is residually \(S\). The context for this result is the following. \textit{M. W. Liebeck} and \textit{A. Shalev} [in J. Algebra 268, No. 1, 264-285 (2003; see the preceding review Zbl 1034.20025)], proved this theorem under the extra hypothesis that the ranks of the groups in \(S\) are unbounded. Also the corresponding problem where \(S\) is an infinite set of alternating groups was settled by \textit{M. C. Tamburini} and \textit{J. S. Wilson} [in Math. Z. 186, 525-530 (1984; Zbl 0545.20019)]. Some assumption on the infinite set \(S\) of finite simple groups in the theorem is necessary; if \(A\) itself is simple of rank \(r\), then certainly \(S\) will have to contain members of rank at least \(r\). The proof of the theorem is probabilistic in nature.