About the finite groups whose minimal normal subgroups are union of two conjugacy classes exactly (Q1110654)

In any finite group G, the number b of minimal normal subgroups is obviously less than the number k of conjugacy classes of G contained in the socle S(G). G is called a \(\Gamma\)-group if \(b=k-1\). Equivalently, G is a \(\Gamma\)-group if and only if for every \(x\in S(G)\), the class containing x together with the unit element is a minimal normal subgroup of G. In such a group, \(| S(G)|\) is a prime-power \(p^ n\). All \(\Gamma\)-groups in which \(| G/S(G)| =p_ 1...p_ m\), where \(m\leq 3\) and \(p_ 1,...,p_ m\) are not necessarily distinct primes, are determined. The first step is to exclude the case when \(| G/S(G)| =p^ m\) for p odd and \(m>1\). The cases when \(| G/S(G)|\) is p or 4 are easily dealt with, but the list of groups for which \(| G/S(G)| =8\) is already quite a long one. When \(| G/S(G)| =pq\), the easier case is the non-cyclic one. The list of those \(\Gamma\)- groups G for which \(| G/S(G)| =pqr\) is a complicated one. The theorems proved can be used to complete the classification of all finite groups with at most 13 conjugacy classes.