, group theory
, Mathematics Metanode
Given groups G and H, and a homomorphism φ : H --> Aut G (where Aut G is the group of automorphisms of G), the semidirect product G ×φ H is
- the set of elements ( g , h ) with g in G, h in H, and
- the operation ( g , h ) ( g' , h' ) = ( g [φ( h ) ( g' )] , h h' )
The semidirect product forms a group, with identity ( e, e ) and ( g , h ) -1 = ( φ( h-1 ) ( g-1 ) , h-1 ). G is a normal subgroup of G ×φ H. Note that the direct product of G and H is just the semidirect product with φ being the trivial homomorphism.
Given an exact sequence 0 --> G --> P --> H --> 0 with homomorphisms i : G --> P and j : P --> H, we say that the sequence splits if there exists a homomorphism k : H --> P such that j ⋅ k is the identity map on P.
P is isomorphic to the semidirect product G ×φ H if and only if there exists a split exact sequence 0 --> G--> P --> H --> 0.
The semidirect product is very useful in constructing (and deconstructing) groups of certain sizes and with certain properties. For instance, take G and H are finite groups, with p and q elements, respectively, such that p and q are prime, p > q, and q divides p - 1. Then there exists a nontrivial homomorphism φ : H --> Aut G, and we therefore have a semidirect product G ×φ H which is non-abelian and has pq elements.
The most common symbol for the semidirect product is Χ|
or something similar; I don't know how (if possible) to make this symbol appear in HTML