On the equipartitions of finite groups
Let G be a finite group. A partition of G is a collection \(\pi =\{H_ 1,...,H_ n\}\) of proper subgroups of G, none of which is \(\{\) \(1\}\), such that every nonidentity element of G is contained in exactly one \(H_ i\in \pi\). The subgroups \(H_ i\) are said to be the components of \(\pi\). \(\pi\) is an equipartition if all components have the same order. \textit{I. M. Isaacs} proved the following theorem: if a finite group G has an equipartition, then G is a non cyclic p-group with \(\exp (G)=p\) [Pac. J. Math. 49, 109-116 (1973; Zbl 0235.20021)]. From this follows that a finite noncyclic 2-group with equipartition is elementary abelian. If G is a noncyclic p-group of exponent p, we can take as a partition the collection of all subgroups of order p. This equipartition is called the atomic equipartition. A p-group G has submaximal class m-1 if its lower central series \[ G=\nu_ 0(G)>\nu_ 2(G)>...>\nu_{m-1}(G)>\nu_ m(G)=\{1\} \] satisfies (i) m-1\(\geq 3\), (ii) \(| \nu_ i(G):\nu_{i+1}(G)| =p\) \((i=2,...,m-1)\). In this paper, it is proved that the p-groups of submaximal class do not admit equipartitions other than the atomic one (Theorem 3.2). This result is then used to give a complete classification of groups of order \(p^ 6\) admitting non-atomic equipartitions (Theorem 5.1).
scientific article; zbMATH DE number 4173229