nLab
semidirect product group

Definitions

If a group $G$ acts on a group $\Gamma$ (on the left, say) by group automorphism

$$\rho :G\to \mathrm{Aut}(\Gamma ),$$\rho : G \to Aut(\Gamma) \,,

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

$$(\delta ,h)(\gamma ,g)=(\delta \rho (h)(\gamma ),hg)$$(\delta,h)(\gamma,g)= (\delta \rho(h)(\gamma) , h g)

for $\delta ,\gamma \in \Gamma ,h,g\in G$, where ${}^h\gamma$ denotes the result of acting with $h$ on the left on $\gamma$.

This is called in group theory the semidirect product and written $\Gamma ⋊G$.

There is a projection morphism $p:\Gamma ⋊G\to G$ , $(\gamma ,g)\to g$. A section $s$ of this can be identified with a derivation $d$, i.e. $d$ satisfies $d(hg)=(dh){}^h(dg)$.

Interior semidirect products

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

The internal and external concepts are equivalent. In particular, any (external) semidirect product $\Gamma ⋊G$ is an internal semidirect product of the images of $\Gamma$ and $G$ 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 $g\gamma$.

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

Semidirect products of groupoids

It is useful to generalise this to the case $\Gamma$ is a groupoid. This occurs if for example $\Gamma ={\pi }_{1}X$ where $X$ is a (left) $G$-space.

So if $X=\mathrm{Ob}(\Gamma )$, then $\Gamma ⋊G$ has object set $X$ and a morphism $y\to x$ is a pair $(\gamma ,g)$ such that $\gamma :y\to gx$ in $\Gamma$. The composition law is then given again by

$$(\delta ,h)(\gamma ,g)=(\delta {}^h\gamma ,hg)$$(\delta,h)(\gamma,g)= (\delta \, ^h \gamma, h g)

if $(\delta ,h):z\to y$, so that $\delta :z\to hy$ in $\Gamma$.

If $\Gamma$ is a discrete groupoid, and so identified with $X$, then we get $X⋊G$ which is the action groupoid of the action. In this case the projection $p:X⋊G\to G$ is a covering morphism of groupoids, i.e. any $g\in G$ has a unique lifting with given initial point. Note that if $Y\to X$ is a covering map of spaces, then the induced morphism of fundamental groupoids is a covering morphism of groupoids. If $q:H\to {\pi }_{1}X$ is a covering morphism of groupoids, and $X$ admits a universal covering map, then there is a topology on $Y=\mathrm{Ob}(H)$ such that $H\cong {\pi }_{1}Y$. In this way, the category of covering maps of $X$ is equivalent to the category of covering morphisms of ${\pi }_{1}X$.

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

$$(\Gamma ⋊G)/N$$(\Gamma \rtimes \, G)/N

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

Properties

As split group extensions

Semidirect product groups $A{⋊}_{\rho }G$ are precisely the split group extensions of $G$ by $A$. See at group extension – split extensions and semidirect product groups.

Examples

The automorphisms on the circle group

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

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

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

$$U(1)\stackrel{}{\hookrightarrow }U(1){⋊}_{{\rho }_{\mathrm{aut}}}{\mathbb{Z}}_2\to {\mathbb{Z}}_2$$U(1) \stackrel{}{\hookrightarrow} U(1) \rtimes_{\rho_{aut}} \mathbb{Z}_2 \to \mathbb{Z}_2

where the group operation is given by

$$(c_1\mathrm{mod}\mathbb{Z},{\sigma }_1)\cdot (c_2\mathrm{mod}\mathbb{Z},{\sigma }_2)=(c_1+{\sigma }_1(c_2)\mathrm{mod}\mathbb{Z},{\sigma }_1+{\sigma }_2).$$(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

• Wikipedia, Semidirect product

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.