- #1

- 1,456

- 44

## Homework Statement

Prove that the quaternion group ##Q_8## is not isomorphic to a semi-direct product ##H\rtimes_\rho K## for non-trivial groups ##H## and ##K##.

## Homework Equations

## The Attempt at a Solution

My idea is to look at the subgroups of ##Q_8## and to show that the intersection of any two nontrivial subgroups is nontrivial. Is this approach valid? My only problem I think is showing that if ##Q_8## were to be isomorphic to a semi-direct product, then ##H## and ##K## must be subgroups of ##Q_8##. Is this true in general?

Last edited: