Normal coverings of solvable groups.
If \(G\) is a noncyclic finite group, then \(G\) can be obtained as the union of some proper subgroups. The smallest integer \(n\) such that \(G\) is the union of \(n\) proper subgroups is denoted by \(\sigma(G)\). This number has been investigated by several authors (see references of the paper). It is clear that there are some restrictions for \(\sigma(G)\) and, in the solvable case, these are quite tight. For instance, \(\sigma(G)=n\) for a solvable group \(G\) if and only if \(n-1\) is a prime power. In this paper, the authors consider a variation on this problem. For noncyclic groups \(G\), the authors define \(\gamma(G)\) to be the smallest integer \(n\) for which there exist \(n\) proper subgroups such that \(G\) is the union of all conjugates of those subgroups. It is not hard to see that \(\gamma(G)\) cannot be 1, and it is shown in this paper that every other positive integer occurs as \(\gamma(G)\), and in fact, \(G\) can be taken to be solvable.
scientific article; zbMATH DE number 6006283
scientific article