H$ .

Asked Jul 15 '18 at 15:07

Active Jul 16 '18 at 11:50

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Let $G$ be a finite group with a normal subgroup $H$ . Then precisely when $G$ is isomorphic to $ H × G/H$ .

I was trying to directly construct an isomorphism with trial and error between $G$ and $ H × G/H$ and my hope was to find appropriate conditions on $G$ and $H$ along the way.But so far not able to make any notable progress.

I was trying different onto homomorphisms from $G\to H$ and $G\to G/H$ , and then use these homomorphisms as the first and the second component of the desired isomorphism between $G$ and $ H × G/H$. I was trying to see how those homomorphisms should relate to each other in order to do the job and what constrain they put on $G$ and $H$ ?

Any progress in this situation will be help full to me .Thanks

edited Jul 16 '18 at 11:38

asked Jul 15 '18 at 15:07

bumba

It is not very clear to me, what are you asking. Please give more details. – peterh Jul 15 '18 at 15:44
1

This is a very broad question. One characterization is when there is a projection $\pi:G\to H$ which is the identity on $H$. – Ashwin Trisal Jul 15 '18 at 15:47
A necessary (but not sufficient) condition is that $H$ is in the center of $G$. – Andreas Blass Jul 15 '18 at 17:45

Let $G$ be a finite group with a normal subgroup $H$ such that $G/H$ has prime order . Then precisely when $G$ is isomorphic to $ H × G/H$ .

0 Answers0