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, …

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 …

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.

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 …

Oct 25, 2016 · 5 I am having a hard time understanding when to use each different subgroup test and if given at any case, its sufficient to use either to proof that the given subgroup is truly a …

The commutator subgroup is generated by commutators. Show that the property of "being a commutator" is invariant under conjuation (in fact it is invariant under all automorphisms).

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 …

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 …

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 …