[GAP Forum] Semidirect product
Quantum.Algebra
quantum.algebra at algebra.at
Sun Aug 31 16:18:15 BST 2014
Dear GAP forum,
are the following versions of semidirect products isomorphic?
Suppose N and S are groups, and h in Hom(S,Aut(N))
is a homomorphism from S to the automorphism group Aut(N) of N.
Then we can define group operations on the cartesian product
N*S = { (n,s) | n in N, s in S } in two ways:
Right-sided version (as usual):
(n,s)*(m,t) := (n*[h(s)](m), s*t), where h(s) in Aut(N),
with inverse elements (n,s)^{-1} = ([h(t^{-1})](s^{-1}),t^{-1}).
Left-sided version (rather unusual):
(n,s)*(m,t) := ([h(t^{-1})](n)*m, s*t), where h(t^{-1}) in Aut(N),
with inverse elements (n,s)^{-1} = ([h(t)](s^{-1}),t^{-1}).
The operations are associative, with neutral element (1,1).
For both group operations,
N*1 is a normal subgroup of N*S
and 1*S is a subgroup of N*S.
Does the assigned homomorphism h determine
a semidirect product structure on N*S uniquely ?
Daniel C. Mayer
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