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Schreier type theorems for bicrossed products

Ana Agore, Gigel Militaru (2012)

Open Mathematics

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...

Several quantitative characterizations of some specific groups

A. Mohammadzadeh, Ali Reza Moghaddamfar (2017)

Commentationes Mathematicae Universitatis Carolinae

Let G be a finite group and let π ( G ) = { p 1 , p 2 , ... , p k } be the set of prime divisors of | G | for which p 1 < p 2 < ⋯ < p k . The Gruenberg-Kegel graph of G , denoted GK ( G ) , is defined as follows: its vertex set is π ( G ) and two different vertices p i and p j are adjacent by an edge if and only if G contains an element of order p i p j . The degree of a vertex p i in GK ( G ) is denoted by d G ( p i ) and the k -tuple D ( G ) = ( d G ( p 1 ) , d G ( p 2 ) , ... , d G ( p k ) ) is said to be the degree pattern of G . Moreover, if ω ⊆ π ( G ) is the vertex set of a connected component of GK ( G ) , then the largest ω -number which divides | G | , is said to be an...

Simple group contain minimal simple groups.

Michael J. J. Barry, Michael B. Ward (1997)

Publicacions Matemàtiques

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.

Small-sum pairs in abelian groups

Reza Akhtar, Paul Larson (2010)

Journal de Théorie des Nombres de Bordeaux

Let G be an abelian group and A , B two subsets of equal size k such that A + B and A + A both have size 2 k - 1 . Answering a question of Bihani and Jin, we prove that if A + B is aperiodic or if there exist elements a ∈ A and b ∈ B such that a + b has a unique expression as an element of A + B and a + a has a unique expression as an element of A + A , then A is a translate of B . We also give an explicit description of the various counterexamples which arise when neither condition holds.

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