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group theory

# Content

## Definitions

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

$\rho : G \to Aut(\Gamma) \,,$

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

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

for $\delta, \gamma \in \Gamma,\; h,g \in G$. For the rest of this page, $^h \gamma \coloneqq \rho(h)(\gamma)$ denotes the result of acting with $h$ on the left on $\gamma$.

If the twist is trivial, then this reduces to just the direct product group construction, whence the name.

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

### 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 $\Gamma \rtimes G$ 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 $(g,h) \mapsto g^h \coloneqq {}^{h^{-1}}g$ and vice versa.

Consider the category $GrpActions$ with

• objects$(G,\Gamma,\rho)$ where $G$ and $\Gamma$ are groups and $\rho: G \to \Aut(\Gamma)$ is a group homomorphisms, and whose

• morphisms$(G,\Gamma,\rho) \to (G',\Gamma',\rho')$ are $G$-equivariant pairs of morphisms $f: G \to G'$ and $h: \Gamma \to \Gamma'$, i.e. such that $h(\rho(g)(\gamma)) = \rho(f(g))(h(\gamma))$ for all $g \in G$ and $\gamma \in \Gamma$.

There is a forgetful functor $U: Arr(Grp) \to GrpActions$ from the arrow category of Grp, sending a group homomorphism $f\colon G \to \Gamma$ to $(G,\Gamma,\rho)$ where $\rho: G \to Aut(\Gamma)$ is given by conjugation, i.e. $\rho(g)(\gamma) = g \gamma g^{-1}$.

Now, its left adjoint functor $GrpActions \to Arr(Grp)$ maps $(G,\Gamma,\rho)$ to the inclusion $G \hookrightarrow \Gamma \rtimes_\rho G$.

### As a Grothendieck construction

Writing $\mathbb{B} G$ for the category with a single object $\ast$ and the group $G$ as its hom set (i.e. the delooping groupoid of $G$), define a functor $F \colon \mathbb{B}G \to$ Cat to send that single object to the delooping groupoid of $\Gamma$, i.e. $* \mapsto \mathbb{B}\Gamma$ and to send the morphisms $G \to Aut(\Gamma)$ according to the given action of $G$ on $\Gamma$.

Then the delooping of the semidirect product group $\Gamma \rtimes G$ arises as the Grothendieck construction of this functor:

$\mathbb{B}( \Gamma \rtimes G) \;\simeq\; \int_{\mathbb{B}G}F$

### 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 $X=Ob(\Gamma)$, then $\Gamma \rtimes \, G$ has object set $X$ and a morphism $y \to x$ is a pair $(\gamma,g)$ such that $\gamma: y \to g x$ in $\Gamma$. The composition law is then given again by

$(\delta,h)(\gamma,g)= (\delta \, ^h \gamma, h g)$

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

If $\Gamma$ is a discrete groupoid, and so identified with $X$, then we get $X \rtimes \, G$ which is the action groupoid of the action. In this case the projection $p: X \rtimes \, G \to G$ 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 $Y=Ob(H)$ 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

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

where $N$ is the normal closure in $\Gamma \rtimes \, G$ 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 $A \rtimes_\rho G$ 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 $U(1) = \mathbb{R}/\mathbb{Z}$ the circle group, the automorphism group is

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

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

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

where the group operation is given by

$(c_1 \; mod \; \mathbb{Z}, \sigma_1) \cdot (c_2\; mod \; \mathbb{Z}, \sigma_2) = (c_1 + \sigma_1(c_2) \; mod \; \mathbb{Z}, \sigma_1 + \sigma_2) \,.$

## 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.

Last revised on August 30, 2021 at 10:16:50. See the history of this page for a list of all contributions to it.