If a group $G$ acts on a group $\Gamma$ (on the left, say) by group automorphism
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 $\rho$:
for $\delta, \gamma \in \Gamma,\; h,g \in G$, where $^h \gamma$ 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 $d(h g) = (d h) \,^h (d g)$.
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.
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 $G \hookrightarrow \Gamma \rtimes_\rho G$.
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:
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
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
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).
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.
For $U(1) = \mathbb{R}/\mathbb{Z}$ the circle group, the automorphism group is
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
where the group operation is given by
A general survey is in
Lecture notes include
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 April 21, 2021 at 12:37:39. See the history of this page for a list of all contributions to it.