External semidirect product

Definition

Definition with the left action convention

Suppose $N$ is a group and $H$ is a group acting on $N$ ; in other words, there is a group homomorphism $\rho :H\to \operatorname {Aut} (N)$ , from $H$ to the automorphism group of $N$ . The external semidirect product $G$ of $N$ and $H$ , denoted $N\rtimes H$ is, as a set, the Cartesian product $N\times H$ , with multiplication given by the rule:

$\!(a,b)(a',b')=(a(\rho (b)(a')),bb')$

Writing the action $\rho (b)a'=b\cdot a'$ , we get:

$(a,b)(a',b')=(a(b\cdot a'),bb')$

The way multiplication is defined, it turns out that:

$N$ embeds as a normal subgroup of $G$ (via $a\mapsto (a,e)$ ) and $H$ embeds as a subgroup via $b\mapsto (e,b)$ . The two subgroups are permutable complements, hence the external semidirect product is the same as an internal semidirect product once we identify $N$ and $H$ with their images in $G$ . In particular, the image of $N$ is a complemented normal subgroup in $G$ and the image of $H$ is a retract of $G$ .
The action of the image of $H$ , on the image of $N$ , via conjugation in $G$ , is the same as the abstract action that we started with.

Example

We will construct the smallest odd-order non-abelian group, the Frobenius group of order 21, as a semidirect product $\mathbb {Z} _{7}\rtimes \mathbb {Z} _{3}$ .

We need to find a group homomorphism $\rho :\mathbb {Z} _{3}\to \operatorname {Aut} (\mathbb {Z} _{7})\cong \mathbb {Z} _{6}$ .

Consider the map $x\mapsto (y\mapsto 2^{x}y)$

Here, $x$ is an element of the integers mod $3$ , and $y$ is an element of the integers mod $7$ .

This is a homomorphism, since $\rho (xx')=y\mapsto 2^{xx'}y=(y\mapsto 2^{x}y)\circ (y\mapsto 2^{x'}y)=\rho (x)\rho (x')$ . Indeed, $\rho (1)=y\mapsto y$ , the identity element of $\mathrm {Aut} (\mathbb {Z} _{7})$ .

Then $G=\mathbb {Z} _{7}\rtimes _{\rho }\mathbb {Z} _{3}$ is non-abelian since $(0,1)*(1,0)=(2,1)$ , $(1,0)*(0,1)=(1,1)$ .

Hence we have constructed a non-abelian group of order $21$ .

The classification of groups of order 21 says that there is only one non-abelian group of order $21$ , the Frobenius group, hence this is the Frobenius group.

Comments

Case of abelian normal subgroup

In the special case where $N$ is an abelian group and the binary operation of $N$ is denoted additively, the multiplication rule for $G$ can be written as:

$\!(a,b)(a',b')=(a+(b\cdot a'),bb')$

This notation comes up in the study of the second cohomology group.

Case of trivial action

The external semidirect product becomes an external direct product when the action of $H$ on $N$ is trivial.

Related notions

Related notions for groups

External direct product (corresponding internal notion: internal direct product, equivalence)
External wreath product (corresponding internal notion: internal wreath product)