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We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H α⋈β G ≅H α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then...
Let be a finite group and let be the set of prime divisors of for which . The Gruenberg-Kegel graph of , denoted , is defined as follows: its vertex set is and two different vertices and are adjacent by an edge if and only if contains an element of order . The degree of a vertex in is denoted by and the -tuple is said to be the degree pattern of . Moreover, if is the vertex set of a connected component of , then the largest -number which divides , is said to be an...
It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.
Let be an abelian group and two subsets of equal size such that and both have size . Answering a question of Bihani and Jin, we prove that if is aperiodic or if there exist elements and such that has a unique expression as an element of and has a unique expression as an element of , then is a translate of . We also give an explicit description of the various counterexamples which arise when neither condition holds.
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