Show that the subgroup $K = \{g^{-1}h^{-1} g h | g, h \in G\}$ is a normal subgroup of a group G

Question

A subgroup N of G is called normal if for every n ∈ N and g ∈ G, gng−1 ∈ N.

a) Given an arbitrary subgroup H of a group G, is there a largest subgroup of G containing H as a normal subgroup? Prove your answer.

b) Show that the subgroup K = h{g−1h−1gh | g, h ∈ G}i is a normal subgroup of a group G

You say that $K$ is a subgroup, but the set you say is equal to $K$ is not necessarily a subgroup. — ancient mathematician, Apr 08 '22 at 08:37
As written, the statement is wrong. The set of commutators does not necessarily form a subgroup, but only generates one — ahulpke, Apr 08 '22 at 12:04

score 2 · Answer 1 · answered Apr 08 '22 at 12:09

2

(Unfortnutly, my reputation is not yet high enough to just comment this.)

Take $k\in K$ and $g\in G$, then $g^{-1}kg=k\underbrace{k^{-1}g^{-1}kg}_{\in K}\in K$.

answered Apr 08 '22 at 12:09

1 Answers1