Semi-direct product

From Encyclopedia of Mathematics
Revision as of 16:55, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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.


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


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

How to Cite This Entry:
Semi-direct product. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article