Set-direct factorizations of groups

Levy, Dan and Maróti, Attila (2018) Set-direct factorizations of groups. JOURNAL OF ALGEBRA, 516. pp. 414-436. ISSN 0021-8693

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Abstract

We consider factorizations G = XY where G is a general group, X and Y are normal subsets of G and any g E G has a unique representation g= xy with x is an element of X and y is an element of Y. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group G has such a factorization if and only if G is a central product of < X > and < Y > and the central subgroup < X >boolean AND < Y > satisfies certain abelian factorization conditions. We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization. (C) 2018 Elsevier Inc. All rights reserved.

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