# Homework Help: Internal semidirect product and roots of unity

1. Sep 3, 2014

### mahler1

1. The problem statement, all variables and given/known data

Let $G=G_{12}$, $H_1=G_3$, $H_2=G_2$. Decide if there are groups $K_1$, $K_2$ such that $G$ can be expressed as the internal semidirect product of $H_i$ and $K_i$.

3. The attempt at a solution

Suppose I can express $G_{12}$ as an internal semidirect product between $G_3$ and $K$, with $K$ subgroups. Then it must be $G_3 \cap K={1}$ and $G_12=G_3.K$. I know that for $n,m$, $G_{n} \cap G_{m}={1}$ if and only if $(n:m)=1$. Taking this condition into account and the fact that $K$ has to be a subgroup of $G_{12}$, the candidates for $K$ would be $G_i$ with $i \in \{2,4\}$. I am not so sure how to realize if one of these subgroups could work. The case $H_2=G_2$ is analogous, I would appreciate suggestions.

Last edited: Sep 3, 2014