Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups.
I know that the only two possible subgroups would be the subgroups of order 4 and 2. I am thinking since D4 is not commutative I can get a contradicition this way, but I am not sure how to do it. Any help would be welcomed
As u said, the possible proper subgroups of D4 are all abelian (groups of order 2, 4). If D4 are the internal direct product of its nontrivial proper subgroups, then D4 becomes abelian. Contradiction.
As Drexel28 said,
, where and the assoicated action is for all h in and x in such that .