# Naming the groups in a semidirect product

A group $$G$$ has the structure of an inner semidirect product when it can be reconstructed from two of its subgroups: one, often written $$N \subset G$$, is a normal subgroup, and the other one, $$H \subset G$$ is a complement subgroup such that $$G = \{ n h : n \in N, h \in H \}$$.

In all the materials I've read, those subgroups do not have a particular name. However, I want to give them explicit names in a MATLAB library I'm writing. Are you aware of a particular terminology here? For the inner semidirect product, I could use "normal subgroup" and "complement subgroup".

However, I'm dealing with outer semidirect products, where a group $$G$$ is composed of pairs $$(n,h) \in N \times H$$ with a particular group operation. How could I call the groups $$N$$ and $$H$$ involved there?

• But "factor group" is often used as a synonym for "quotient group", so there could be a danger of cofusion. Perhaps "factor subgroup" would be better? – Derek Holt Mar 29 at 13:53
• @DerekHolt I agree. Also, the "normal complement" should refer to the normal subgroup and not the other one! So I've deleted my comment. – user1729 Mar 29 at 14:06
• Oh, I just remembered where I came across the term "normal complement"! If $G=N\rtimes H$ then $H$ is a retract of $G$ with normal complement $N$. – user1729 Mar 29 at 14:08

If $$G=N\rtimes H$$ then $$H$$ is a retract of $$G$$ with normal complement $$N$$.