ACTA issues

Covering systems of direct products of elementary $p$-groups are regular

István Kovács

Acta Sci. Math. (Szeged) 79:1-2(2013), 31-41

Abstract. A covering system of a finite group $G$ is a set ${\eufm S}$ of pairs of its subgroups, ${\eufm S}= \{ (L_1,M_1), \ldots, (L_n,M_n) \},$ which satisfies the following axioms: $M_i < L_i$ for every $i \in\{1,\ldots,n\},$ $\bigcup_{i=1}^n(L_i \setminus M_i) = G \setminus\{e\},$ and $|G| = \prod_{i=1}^n |L_i : M_i|,$ where $e$ is the identity element of $G$. The covering system ${\eufm S}$ is said to be regular if $L_i=G$ for some $i \in\{1,\ldots,n\} $. In this paper we show that every covering system of every direct product of elementary $p$-groups is regular.

AMS Subject Classification (1991): 20K01, 20K27

Keyword(s): abelian group, covering system, group character

Received August 23, 2011, and in revised form January 9, 2013. (Registered under 41/2011.)