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 …

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 …

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

a subgroup of G G? If G G is abelian, then this is clearly the case. I believe I have shown that if H H and K K are both normal subgroups of G G, then HK H K is a subgroup. Are there any more …

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, …

Dec 23, 2019 · Hint the First: for arbitrary subgroups H and K, HK is a subgroup if and only if HK = KH as sets. Hint the Second: if K is normal, then for each h ∈ H, hK = Kh. As to your work: …

The subgroup generated by 1 and 1/2 is generated by 1/2 alone. There are two cases: there is a smallest positive element in this subgroup (which you must show generates the group), or …

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 ∈ …