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 …

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...

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

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

Nov 11, 2021 · 4 I think it means most of the time that the subgroup H H is just not equal to G G, see here. In this sense, if G G is a non-trivial group, the subgroup {e} {e} is a proper subgroup, …

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 …

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 ...

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...

Mar 7, 2017 · The definition of a normal subgroup is that gN = Ng g N = N g for all g ∈ G g ∈ G, or equivalently gNg−1 = N g N g 1 = N for all g ∈ G g ∈ G. Note the following pitfall: for a fixed g ∈ …

So if you have two subgroups H, K ≤ G H, K ≤ G such that H ≰ K H ≰ K and K ≰ H K ≰ H, the join H, K H, K is another subgroup of G G. Chapter 2, Section 5 in Dummit and Foote has a good …