Subgroups of a direct product - Mathematics Stack Exchange
Until recently, I believed that a subgroup of a direct product was the direct product of subgroups. Obviously, there exists a trivial counterexample to this statement. I have a question regarding...
What is the difference between a Subgroup and a subset?
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 …
Subgroup generated by a set - Mathematics Stack Exchange
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 …
What exactly a proper subgroup means? - Mathematics Stack …
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) …
group theory - Subgroup criterion. - Mathematics Stack Exchange
Aug 10, 2012 · I've been reading some stuff about algebra in my free time, and I think I understand most of the stuff but I'm having trouble with the exercises. Specifically, the …
A normal subgroup is the union of conjugacy classes.
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 …
$Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic …
Suppose $Q_8$ is isomorphic to subgroup of $S_n$ with $n\leq 7.$ Then it should come from a group action of $Q_8$ on a set of cardinality at most 7. Suppose $Q_8$ acts on a set $A$ with …
Are normal subgroups transitive? - Mathematics Stack Exchange
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 …
$HK$ is a subgroup of $G$ if and only if $HK = KH$
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 ...
Understanding how to prove when a subset is a subgroup
Understanding how to prove when a subset is a subgroup Ask Question Asked 9 years, 1 month ago Modified 4 years ago