See

group,

group theory,

Mathematics Metanode.

**Definition**

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.

**Definition**

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`.

**Theorem**

`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.