Treffer: On groups with conjugacy classes of distinct sizes: On groups with conjugacy classes of distinct sizes.

We say that a finite group \(G\) is an ah-group if any two distinct conjugacy classes of \(G\) have different sizes. (The ah-group name comes from the phrase anti-homogeneous.) An ah-group is easily seen to be rational, meaning that all its characters are rational valued. This fact alone greatly restricts the structure of an ah-group. However, it is actually conjectured that any non-trivial ah-group is isomorphic to the symmetric group \(S_3\). This seemingly elementary problem has resisted attempts at solution and requires at the bare minimum the classification of finite simple groups to get started. The conjecture is known to be true in the solvable case, but even here the arguments are not entirely straightforward. The authors investigate the structure of an ah-group in the non-solvable case in the paper under review. Their main result is the following. Let \(G\) be an ah-group. Then if the non-Abelian socle of \(G\) is non-trivial, it is isomorphic to one of the following: \(A_5^a\), for \(1\leq a\leq 5\), \(a\neq 2\); \(A_8\); \(\text{PSL}(3,4)^e\), for \(1\leq e\leq 10\); \(A_5\times\text{PSL}(3,4)^e\), for \(1\leq e\leq 10\). They also show that if \(G\) is an ah-group of minimal size which is not isomorphic to \(S_3\), the non-Abelian socle of \(G\) is either trivial or isomorphic to \(A_5^a\), for \(3\leq a\leq 5\), or to \(\text{PSL}(3,4)^e\), for \(1\leq e\leq 10\). The proofs involve considerable amounts of computation, details of which are given in appendices.