External semidirect product of semigroup and group

From Groupprops
Revision as of 00:45, 21 April 2010 by Vipul (talk | contribs) (Created page with '==Definition== ===Definition with the left action convention=== Suppose <math>S</math> is a semigroup and <math>H</math> is a group along with a [[homomorphism of groups|ho…')
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Definition

Definition with the left action convention

Suppose S is a semigroup and H is a group along with a homomorphism \rho:H \to \operatorname{Aut}(S) (the automorphism group of S). The external semidirect product of S by H with respect to \rho, denoted S \rtimes H, is defined as the following semigroup T. As a set T = S \times H, with the multiplication given by:

\! (a,b)(a',b') = (a (\rho(b)(a')), bb')

If we denote \rho(b)(a') by b \cdot a', this can be rewritten as:

\! (a,b)(a',b') = (a (b \cdot a'), bb')