There are different notions of homology and cohomology that can be defined for a group with an action another by automorphisms. In this paper we address three natural questions arise in context. Namely, the relation these usual (co)homology semidirect product, interpretation first as some kind abelianization classification (invariant) extensions.

The method of constructing the generalized dihedral group as a semidirect product an abelian and Z2 integers modulo 2 is extended to case gyrogroups. This leads study new class gyrogroups, which includes groups special case. In this article, we show that any dihedralizable gyrogroup can be enlarged dihedralized gyrogroup. Then, establish algebraic properties gyrogroups well combinatorial their ...

An algorithm for calculating a set of generators of representative 2-cocycles on semidirect product of finite abelian groups is constructed, in light of the theory over cocyclic matrices developed by Horadam and de Launey in [7,8]. The method involves some homological perturbation techniques [3,1], in the homological correspondent to the work which Grabmeier and Lambe described in [12] from the...

This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided s...

The categorical definition of semidirect products was introduced by D. Bourn and G. Janelidze in [2], where they proved that, in the category of groups, this notion coincides with the classical one. A characterization of pointed categories with categorical semidirect products was given in [3]. The existence of such products imply, in particular, that the category is protomodular, i.e. the Split...

We describe all left braces of size 8p for an odd prime p?3,7 and validate the number given by Bardakov, Neschadim Yadav in [2]. give a characterization isomorphism classes semidirect product then description is done first describing 8, as conjugacy regular subgroups corresponding holomorph, checking how many non isomorphic are obtained from each one them.

We use the gluing construction introduced by Jia Huang to explore rings of invariants for a range modular representations. construct generating sets maximal parabolic subgroups finite symplectic group and their common Sylow p -subgroup. also investigate singular classical groups. introduce this compute invariant field fractions thin faithful representations semidirect products determine minimum...

For a given Jacobi-Jordan algebra A and vector space V over field k , non-abelian cohomological type object H 2 ( ) is constructed: it classifies all algebras containing as subalgebra of codimension equal to dim . Any such isomorphic so-called unified product ♮ Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed) ⋈ associated two special case product. Several examples...