Vol. 50, No. 1, 1974

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Isomorphisms of group extensions

Kung-Wei Yang

Vol. 50 (1974), No. 1, 299–304
Abstract

Let 0 G E Π 1 and 0 G E′→ Π 1 be two crossed product extensions given by the crossed product groups E = [G,φ,f,Π] and E= [G,φ,f,Π] respectively. A homomorphism Γ : E Eis stabilizing if the diagram

0  −→   G  − →   E   −→   Π  −→   1
∥       ↓ Γ       ∥
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
↓ τ      ↓ Φ′       ↓ σ
0  −→   G   −→    E   −→   Π   − →  1

commutative, given (σ,τ) AutΠ × AutG.

Mathematical Subject Classification
Primary: 20F25
Milestones
Received: 28 September 1972
Published: 1 January 1974
Authors
Kung-Wei Yang