0

Let $A$ and $B$ be two groups of co-prime orders. Let $G=A \times B$. Then I want to find all possible normal subgroups of $G$.

It can be shown that it $A'$ and $B'$ are normal subgroups of $A$ and $B$, respectively, then $A'\times B'$ is a normal subgroup of $G$. Is there any other subgroup of $G$ that is not of the form of direct product of normal subgroups?

I know that it the orders of $A$ and $B$ are not co-prime then there exist normal subgroups that are not direct product.

PAMG
  • 4,440
  • In general, see this post. There is the Lemma by Goursat. – Dietrich Burde Sep 10 '22 at 12:24
  • 1
    @DietrichBurde Is this a good duplicate? This post concerns groups of relatively prime order...the examples presented in the other do not satisfy that condition. – lulu Sep 10 '22 at 12:29
  • @lulu I see. I have not seen the last sentence first. So I have reopened the question (although I think it is known here). So for non-coprime orders there are counterexamples given here. – Dietrich Burde Sep 10 '22 at 12:31

0 Answers0