The degree and index of a finite group
For a finite group \(G\), denote by \(\delta(G)\) the least natural \(n\) such that \(G\) has a faithful permutation representation of degree \(n\) and by \(i(G)\) the minimal index of a proper subgroup of \(G\). The author proves that \(i(G)=\delta(G)\) iff \(G\) is simple (Theorem 1) and classifies all non-simple groups with \(2i(G)>\delta(G)\) (Theorem 6). A lot of similar results is obtained, for example, the following Proposition 11: Let \(p>2\) be the least prime divisor of \(G\). If \(\delta(G)>|G|/p^2\) then \(G\) has an ordered Sylow tower of the same type as supersolvable groups have.
scientific article; zbMATH DE number 1295437
scientific article