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

Authors : Mira BHARGAVA

Pages : 83-89

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Publication Date : 2007-12-01

Article Type : Research Paper

Abstract :We prove that, for any positive integer n, there exists a minimal finite set Sinsert ignore into journalissuearticles values(n); of finite groups such that: a group G is the union of n of its proper subgroups insert ignore into journalissuearticles values(but not the union of fewer than n proper subgroups); if and only if G has a quotient isomorphic to some group K ∈ Sinsert ignore into journalissuearticles values(n);. We prove, furthermore, that such a minimal finite set Sinsert ignore into journalissuearticles values(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”.
Keywords : finite groups, unions of subgroups, unions of cosets, coverings of groups, minimal covers