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 …

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 …

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 …

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 ...

Understanding how to prove when a subset is a subgroup Ask Question Asked 9 years, 3 months ago Modified 4 years, 2 months ago

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 …

Apr 23, 2016 · As others said subgroup has all the properties of Group. But conjugacy classes are just the set, but created with conjugacy and are equivalence relation. Intuitively conjugacy is, looking the …

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 …

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...