Isomorphisms of group extensions
1974; Mathematical Sciences Publishers; Volume: 50; Issue: 1 Linguagem: Inglês
10.2140/pjm.1974.50.299
ISSN1945-5844
Autores Tópico(s)Advanced Topics in Algebra
ResumoTo my parentsLet 0->G-> E-* 77->1 and 0->G-^E'-*Π->l be two crossed product extensions given by the crossed product groups E = [G, φ, f, 77] and E' = [G, φ', /', 77] respectively.A homomorphism Γ: E' -> E' is stabilizing if the diagram 0 >(? >E >Π >1 0 > G > E' > Π > 1 commutes.In this paper, a necessary and sufficient condition for the existence of a stabilizing homomorphism (hence isomorphism) between any two crossed product extensions is obtained.The result is applied to obtain a necessary and sufficient condition for the existence of an automorphism φ: E -> E making the diagram 0 >G >E >Π >1 ϊ φ l σ 0 >G >E >Π >l commutative, given (σ, τ) 6 Aut 77 x Aut G.NOTATION.In general, we use the notation in [3].Throughout the paper, G and 77 denote two fixed groups.G will be written in additive notation and 77 in multiplicative notation.Aut G, Out G, and ZG are the automorphism group, the outer automorphism group, and the center of G, respectively.For any element a e G, μ(a) denotes the inner automorphism μ(a)(g) = a + g -a given by conjugation with α.When X is a group the natural image of an element xe X in a quotient group of X is denoted x.When ψ is a map, φ denotes the map φ(x) = φ(x).Given groups G, 77, and functions φ\ 77 -> Aut G, /: 77 x 77 -> G satisfying
Referência(s)