In this paper, we study an important well-known structure operation, namely, the semidirect product (or split extensions) which is a very useful tool to structure certain kinds of groups. More precisely, we study the isomorphism problem for semidirect products and then we determine how isomorphism of semidirect products and conjugacy of the images of the corresponding actions are related. As an application, for two positive integers [Formula: see text] and [Formula: see text], we compute the number of upper isomorphism classes of split extensions of an elementary abelian [Formula: see text]-group of order [Formula: see text] by an elementary abelian [Formula: see text]-group of order [Formula: see text]. Furthermore, we deal with split extensions where the kernel is a non-abelian [Formula: see text]-group of nilpotency class two and the quotient is an elementary abelian [Formula: see text]-group.
Let [Formula: see text] be a group acting on an abelian group [Formula: see text] via a homomorphism [Formula: see text] and let a 2-cocycle [Formula: see text]. By Schreier’s theorem, the pair [Formula: see text] determines a group [Formula: see text] which can arise as a non-split extension of [Formula: see text] by [Formula: see text], denoted by [Formula: see text] and called the perturbed semidirect product of [Formula: see text] by [Formula: see text] under [Formula: see text]. In this paper, we classify the perturbed semidirect products and give some of their properties. Furthermore, we find necessary and sufficient conditions for two perturbed semidirect products to be isomorphic.
Let p be a prime number and F_p a finite field of order p. Let GL_n(F_p) denote the general linear group and let U_n denote the unitriangular group of n x n upper triangular matrices with ones on the diagonal, over the finite field F_p. This is a finite group of order and a Sylow p-subgroup of GL_n(F_p}. In this work, we characterize some p-subgroups of GL_n(F_p) with respect to a given property.
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