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...

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 …

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) …

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

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...

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 …

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 ...

Jan 6, 2013 · Any finitely generated subgroup of $\mathbb Q$ must be cyclic and, and a subgroup of $\mathbb Q$ is isomorphic to atleast one subgroup of $\mathbb Q$ containing $1$. I think …

Aug 10, 2012 · Subgroup criterion. Ask Question Asked 13 years, 4 months ago Modified 10 years, 5 months ago

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 …