Semidirect products

Definitions

If a group $G$ acts on a group $\Gamma$ on the left, then there is a semidirect product group whose underlying set is $\Gamma \times G$ but whose multiplication is

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 (see for example the Wikipedia entry) 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 particuarl, 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

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

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).

References

• 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.