# Semi-direct product

*of a group by a group *

A group which is the product of its subgroups and , where is normal in and . If is also normal in , then the semi-direct product becomes a direct product. The semi-direct product of two groups and is not uniquely determined. To construct a semi-direct product one should also know which automorphisms of the group are induced by conjugation by elements of . More precisely, if is a semi-direct product, then to each element corresponds an automorphism , which is conjugation by the element :

Here, the correspondence is a homomorphism . Conversely, if and are arbitrary groups, then for any homomorphism there is a unique semi-direct product of the group by the group for which for any . A semi-direct product is a particular case of an extension of a group by a group (cf. Extension of a group); such an extension is called split.

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1 , Chelsea (1960) (Translated from Russian) |

#### Comments

The semi-direct product of by is often denoted by or .

**How to Cite This Entry:**

Semi-direct product.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Semi-direct_product&oldid=11367