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 …

Understanding how to prove when a subset is a subgroup Ask Question Asked 9 years, 3 months ago Modified 4 years, 1 month ago

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 ...

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 every e...

Apr 10, 2013 · How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members ...

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 …

Mar 15, 2021 · The cyclic subgroup of order $4$ contains an element of order $4$, so the only candidates are $r^3$ and $r$.

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) …

So really is it true that if $HK$ is a subgroup of $G$ (or really $HK=G$) does it imply that either $H$ or $K$ normal? I am mainly interested in finite groups and when $G$ factors over $H$ …

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