Jan 31, 2025 · So, if $P$ and $Q$ are Sylow $5$ -subgroup and Sylow $7$ -subgroup of $G$ respectively, then one of the two has to be normal in $G$. Assume $P$ is normal in $G$, that is, the …

Dec 23, 2019 · Hint the First: for arbitrary subgroups $H$ and $K$, $HK$ is a subgroup if and only if $HK=KH$ as sets. Hint the Second: if $K$ is normal, then for each $h\in H$, $hK=Kh$.

A subgroup is a subset which is also a group of its own, in a way compatible with the original group structure. But not every subset is a subgroup. To be a subgroup you need to contain the neutral …

In general, $HK$ is a subgroup if and only if $HK=KH$.

Nov 11, 2021 · The question seems very simple, but it's confusing me as the term 'proper subgroup' has different definations in different reference books. I read in galian(7th edition) that the subgroup of G …

For all the subgroups on the third row from the top, their only proper subgroup is the trivial subgroup, which is trivially normal to $G$, so it doesn't make sense to use any of the subgroups on the third …

Jun 12, 2023 · Let $H$ and $K$ be subgroups of $G$. Prove that $HK$ is a subgroup of $G$ if and only if $HK=KH$. In particular, the condition holds if $hk=kh$ for all $h$ in $H$ and ...

Mar 7, 2017 · – Cloud JR K Sep 22, 2018 at 7:46 How will apply this result, to determine a subgroup of given order is normal, if the class equation of the group is given – sabeelmsk Dec 20, 2019 at 10:00

Jan 3, 2017 · Edit: Written this assuming you've taken a course on group theory. Finding all subgroups of large finite groups is in general a very difficult problem. Usually, I'd start with Lagrange's theorem …

May 15, 2012 · A subgroup generated by a set is defined as (from Wikipedia): More generally, if S is a subset of a group G, then , the subgroup generated by S, is the smallest subgroup of G containing …