Finite groups with few conjugacy classes of subgroups
If \(G\) is a finite group, denote by \(C(G)\) its poset of conjugacy classes of subgroups \(\{[H]\}\), where \([H] \leq [K]\) if \(H \leq K^ g\) for some \(g\in G\). The Dilworth number \(w(G)\) is defined as the maximal cardinal of an antichain of the poset \(C(G)\). The authors study the problem of determining those groups \(G\) with a ``small'' number \(w(G)\). Since for every finite group \(G\) we have \(w(G) \geq | \pi(G)|\), they consider groups \(G\) for which \(w(G) = | \pi(G)| \in \{1,2,3\}\). The case \(w(G) = 1\) is trivial, the groups in question being cyclic \(p\)- groups. In the other cases the key argument is that the structure of the Sylow \(p\)-subgroups of \(G\) is very restricted if \(w(G) = | \pi(G)|\). Using this, the authors quickly dispose of the structure of the nonsolvable groups \(G\) with \(w(G) = | \pi(G)| = 3:\) only \(\text{SL}(2,5)\) and \(\text{PSL}(2,5)\) verify the condition. The bulk of the analysis is devoted to the structure of the solvable groups with \(w(G) = | \pi(G)| \in \{2,3\}\); their list is too long to be included here.
scientific article; zbMATH DE number 94573