Suppose $H$ and $N$ are groups with operations $*_H$ and $*_N$ and $\phi:H\to \Aut(N)$ is a homomorphism to the automorphism group of $N$. Then the set $N\times H$ with the operation
\[ (n_1, h_1)*(n_2, h_2) = (n_1*_N \phi(h_1)(n_2), h_1 *_H h_2)\]
is a group called a **semidirect product** of $H$ and $N$ denoted $N:H$ or $N\rtimes H$.

The group $N:H$ has subgroups \[ N' = \{(n,e)\mid n\in N\} \cong N\] and \[ H' = \{(e,h) \mid h\in H\} \cong H\] satisfying $N'\cap H'=\{(e,e)\}$, $N'H'=N:H$, and $N'$ is a normal subgroup of $N:H$.

Conversely, if a group $G$ has a normal subgroup $N$ and a subgroup $H$ such that $N\cap H=\{e\}$ such that $N\cap H=\{e\}$ and $NH=G$, then $G\cong N:H$ for some function $\phi$. We call $N$ a **semidirect factor** of $G$ in this case.

The semidirect product of $N$ and $H$ fits into a sequence
\[ 1 \to N \to N \rtimes H \to H \to 1, \]
and is distinguished by the existence of a homomorphism $H \to N \rtimes H$ with image $H'$ that splits the projection $N \rtimes H \to H$. We refer to such an extension as **split**, in contrast to other **non-split** extensions.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Jones on 2022-07-20 18:07:11

**Referred to by:**

- columns.gps_groups.semidirect_product
- columns.gps_subgroups.quotient_action_image
- columns.gps_subgroups.quotient_action_kernel
- columns.gps_subgroups.split
- group.dicyclic
- group.name
- group.nonsplit_product
- group.subgroup.complement
- group.wreath_product
- lmfdb/groups/abstract/main.py (line 434)
- lmfdb/groups/abstract/main.py (line 1812)
- lmfdb/groups/abstract/main.py (line 2111)
- lmfdb/groups/abstract/templates/abstract-show-group.html (line 207)

**History:**(expand/hide all)

- 2022-09-03 19:40:13 by John Jones
- 2022-07-20 18:07:11 by John Jones (Reviewed)
- 2022-07-18 18:16:32 by Jennifer Paulhus
- 2022-07-18 17:09:00 by Jennifer Paulhus (Reviewed)
- 2021-10-13 11:43:39 by David Roe
- 2021-09-30 16:47:43 by David Roe
- 2019-05-23 19:35:36 by John Jones

**Differences**(show/hide)