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semidirect product group

Semidirect products

Definitions

If a group GG acts on a group Γ\Gamma (on the left, say) by group automorphism

ρ:GAut(Γ), \rho : G \to Aut(\Gamma) \,,

then there is a semidirect product group whose underlying set is the product Γ×G\Gamma \times G but whose multiplication is twisted by ρ\rho:

(δ,h)(γ,g)=(δρ(h)(γ),hg) (\delta,h)(\gamma,g)= (\delta \rho(h)(\gamma) , h g)

for δ,γΓ,h,gG\delta, \gamma \in \Gamma,\; h,g \in G, where hγ^h \gamma denotes the result of acting with hh on the left on γ\gamma.

This is called in group theory the semidirect product and written ΓG\Gamma \rtimes \, G.

There is a projection morphism p:ΓGGp:\Gamma \rtimes \, G \to G , (γ,g)g(\gamma, g) \to g. A section ss of this can be identified with a derivation dd, i.e. dd satisfies d(hg)=(dh) h(dg)d(h g) = (d h) \,^h (d g).

Interior semidirect products

Let HH be any group. A decomposition of HH as an internal semidirect product consists of a subgroup Γ\Gamma and a normal subgroup GG, such that every element of HH can be written uniquely in the form γg\gamma g, for γΓ\gamma \in \Gamma and gGg \in G.

The internal and external concepts are equivalent. In particular, any (external) semidirect product ΓG\Gamma \rtimes G is an internal semidirect product of the images of Γ\Gamma and GG in it.

Right semidirect products

The definitions above are not symmetric in left and right; since the first definition begins with a left action, we may call it a left semidirect product. Then a right semidirect product is given by an action on the right, or internally by the requirement that every element can be written in the form gγg \gamma.

However, right and left semidirect products are equivalent. Essentially, this is because any left action (h,g) hg(h,g) \mapsto {}^h{g} defines a right action (g,h)g h h 1g(g,h) \mapsto g^h \coloneqq {}^{h^{-1}}g and vice versa.

Semidirect products of groupoids

It is useful to generalise this to the case Γ\Gamma is a groupoid. This occurs if for example Γ=π 1X\Gamma = \pi_1 X where XX is a (left) GG-space.

So if X=Ob(Γ)X=Ob(\Gamma), then ΓG\Gamma \rtimes \, G has object set XX and a morphism yxy \to x is a pair (γ,g)(\gamma,g) such that γ:ygx\gamma: y \to g x in Γ\Gamma. The composition law is then given again by

(δ,h)(γ,g)=(δ hγ,hg)(\delta,h)(\gamma,g)= (\delta \, ^h \gamma, h g)

if (δ,h):zy(\delta, h): z \to y, so that δ:zhy\delta: z \to h y in Γ\Gamma.

If Γ\Gamma is a discrete groupoid, and so identified with XX, then we get XGX \rtimes \, G which is the action groupoid of the action. In this case the projection p:XGGp: X \rtimes \, G \to G is a covering morphism of groupoids, i.e. any gGg \in G has a unique lifting with given initial point. Note that if YXY \to X is a covering map of spaces, then the induced morphism of fundamental groupoids is a covering morphism of groupoids. If q:Hπ 1Xq: H \to \pi_1 X is a covering morphism of groupoids, and XX admits a universal covering map, then there is a topology on Y=Ob(H)Y=Ob(H) such that Hπ 1YH \cong \pi_1 Y. In this way, the category of covering maps of XX is equivalent to the category of covering morphisms of π 1X\pi_1 X.

The utility of the more general construction is that there is notion of orbit groupoid Γ//G\Gamma //G (identify any γ\gamma and gγ^g \gamma) and it is a theorem that the orbit groupoid is isomorphic to the quotient groupoid

(ΓG)/N (\Gamma \rtimes \, G)/N

where NN is the normal closure in ΓG\Gamma \rtimes \, G of all elements (1 x,g)(1_x,g). Details are in the book reference below (but the conventions are not quite the same).

Properties

As split group extensions

Semidirect product groups A ρGA \rtimes_\rho G are precisely the split group extensions of GG by AA. See at group extension – split extensions and semidirect product groups.

Examples

The automorphisms on the circle group

For U(1)=/U(1) = \mathbb{R}/\mathbb{Z} the circle group, the automorphism group is

Aut(U(1)) 2, Aut(U(1)) \simeq \mathbb{Z}_2 \,,

where the nontrivial element in 2\mathbb{Z}_2 acts on \mathbb{R} by multiplication with 1-1. Write ρ aut:U(1)× 2U(1)\rho_{aut} : U(1) \times \mathbb{Z}_2 \to U(1) for the automorphism action. The corresponding semidirect product group is the group extension

U(1)U(1) ρ aut 2 2 U(1) \stackrel{}{\hookrightarrow} U(1) \rtimes_{\rho_{aut}} \mathbb{Z}_2 \to \mathbb{Z}_2

where the group operation is given by

(c 1mod,σ 1)(c 2mod,σ 2)=(c 1+σ 1(c 2)mod,σ 1+σ 2). (c_1 \; mod \; \mathbb{Z}, \sigma_1) \cdot (c_2\; mod \; \mathbb{Z}, \sigma_2) = (c_1 + \sigma_1(c_2) \; mod \; \mathbb{Z}, \sigma_1 + \sigma_2) \,.

References

A general survey is in

Lecture notes include

  • Patrick Morandi, Semidirect products (pdf)

Relevant textbooks include

  • R. Brown, Topology and groupoids, Booksurge 2006.

  • P. J. Higgins and J. Taylor, The Fundamental Groupoid and Homotopy Crossed Complex of an Orbit Space, in K.H. Kamps et al., ed., Category Theory: Proceedings Gummersbach 1981, Springer LNM 962 (1982) 115–122.

Revised on October 10, 2012 12:34:11 by Tim Porter (95.147.236.113)