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?

  • $\begingroup$ 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? $\endgroup$ – Derek Holt Mar 29 at 13:53
  • $\begingroup$ @DerekHolt I agree. Also, the "normal complement" should refer to the normal subgroup and not the other one! So I've deleted my comment. $\endgroup$ – user1729 Mar 29 at 14:06
  • $\begingroup$ 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 $N$. $\endgroup$ – user1729 Mar 29 at 14:08

Some standard terminology is:

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

  • $\begingroup$ Excellent, that's helpful! Though it'd be a bit of a stretch, I may call the groups in the outer construction "retract group" and "normal complement group", as they would be isomorphic to subgroups of the outer construction. $\endgroup$ – Denis Rosset Mar 29 at 14:25

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