FINITENESS CRITERIA FOR COVERINGS OF GROUPS BY FINITELY MANY SUBGROUPS OR COSETS

Abstract

We prove that, for any positive integer n, there exists a minimal finite set S(n) of finite groups such that: a group G is the union of n of its proper subgroups (but not the union of fewer than n proper subgroups) if and only if G has a quotient isomorphic to some group K ∈ S(n). We prove, furthermore, that such a minimal finite set S(n) is in fact unique up to isomorphism of its members. Finally, an analogue of this result can be proved when “subgroups” is replaced more generally by “cosets”.