Generalized Semidirect Product in Group Extensions

Abstract

The case when there exists a homomorphism σ of a group G into Aut(K) of a non‐Abelian group K[σ having at most one image in every coset of Aut(K) with respect to I(K)] is investigated. It is shown that any extension E ∈ ext_σ(G, K) can be obtained as a generalized semidirect product $(\mathrm{GSP}\mathrm{):E=(K̂\tau H)/C\prime }$ , where H belongs to ext_σ(G, C) (the group C being the center of K), the semidirect product of K and H is based on τ which equals $\mathrm{\sigma \circ n}$ (n being the homomorphism of H onto G), and C′ is the antidiagonal of $\mathrm{C\otimes C}$ . The GSP is a natural generalization of the central extensions, it is applicable to most groups in theoretical physics, and it has a suitable form for the derivation of the irreducible representations of E.