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OD-characterization of alternating groupsA p + d

100%
Yang Y., Liu S., Zhang Z.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1090-1098
EN
Let An be an alternating group of degree n. Some authors have proved that A10, A147 and A189 cannot be OD-characterizable. On the other hand, others have shown that A16, A23+4, and A23+5 are OD-characterizable. We will prove that the alternating groups Ap+d except A10, are OD-characterizable, where p is a prime and d is a prime or equals to 4. This result generalizes other results.
2
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Schreier type theorems for bicrossed products

84%
Agore A., Militaru G.
Open Mathematics
|
2012
|
tom 10
|
nr 2
722-739
EN
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 any σ-invariant isomorphism H α⋈β G ≅ H α′⋈β′ G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed products of groups are given.
3
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Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

60%
Bandman T., Garion S., Kunyavskiĭ B.
Open Mathematics
|
2014
|
tom 12
|
nr 2
175-211
EN
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
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