---
tag: 014E1DZQE8
kind: definition
name: Semidirect product
msc: ["20E22"]
defines: 0H3WZXCZNF
uses: [0CWXCWP0Y0, 0HV9Y7WG5R, 0W77CVBCHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Semidirect_product
    locator: Outer semidirect product
  - url: https://en.wikipedia.org/wiki/Semidirect_product
    locator: Inner semidirect product
---

## Definition

Let $N$ and $H$ be [[groups|G5S1M9]], and let $\varphi \colon H \to \operatorname{Aut}(N)$, $h \mapsto \varphi_h$, be a [[group homomorphism|0JYTG5K9Y4]].
The semidirect product $N \rtimes_\varphi H$ is the set $N \times H$ with the operation $(n_1, h_1)(n_2, h_2) = (n_1 \varphi_{h_1}(n_2), h_1 h_2)$.

## Remarks

The identity is $(e, e)$, and the inverse of $(n, h)$ is $(\varphi_{h^{-1}}(n^{-1}), h^{-1})$.
The set $\{(n, e) : n \in N\}$ is a [[normal subgroup|0CB0DX5BVV]] isomorphic to $N$, and $\{(e, h) : h \in H\}$ is a subgroup isomorphic to $H$.
If $\varphi$ is the trivial homomorphism, then $N \rtimes_\varphi H$ is the [[direct product|00HF4RJ5QN]] $N \times H$.
The [[dihedral group|0JZ27DMJ5D]] $D_n$ is $\mathbb{Z}/n\mathbb{Z} \rtimes_\varphi \mathbb{Z}/2\mathbb{Z}$, where the nonzero element of $\mathbb{Z}/2\mathbb{Z}$ acts by $x \mapsto -x$.

## Explanations

### L1 human

In a direct product, the two factors do not interact.
In a semidirect product, the second factor $H$ twists the first factor $N$ when you multiply.
The homomorphism $\varphi$ says exactly how each element of $H$ twists $N$.