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 …

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, since it is …

Therefore, (H, ∗) (H, ∗) is a group, which means that H H is a subgroup of G G. The initial part is clear and makes sense, once you assume H H to be a subgroup.

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 following: Prove that a nonempty

Mar 1, 2023 · I am trying to understand subgroups.I know a given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. So if I where …

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...

Feb 14, 2024 · If the criteria for a subgroup of a group is to be non-empty and have inverse elements for every element in the subgroup, then why don't the subgroups of D4 have inverse elements for each …

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 exposition …

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 there is …

Nov 10, 2014 · As the define goes: A subgroup N N of a group G G is called a normal subgroup if it is invariant under conjugation; that is, for each element n n in N N and each g g in G G, the element …