- #1

AllRelative

- 42

- 2

## Homework Statement

[G,G] is the commutator group.

Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##.

## Homework Equations

## The Attempt at a Solution

In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that ##[G,G]\triangleleft G##.

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I am unsure what ##(H\cap [G,G])## represents.

Is ##(H\cap [G,G])## = ##[H,H]##,

or is it equal to {##[x,y] \in H | x,y \in G##}??

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I began my solution like this:

##\forall [a,b] \in (H\cap [G,G])##, we have that ##a^{-1}b^{-1}ab = e##

##\Rightarrow ab = ba##

something missing

##\Rightarrow H\subseteq Z(G)##.