1. ## commutator subgroup eep!

here's the prob

Prove if K is a normal subgroup of a group G then $\displaystyle K'\trianglelefteq G$ where $\displaystyle K'=[K,K] =\langle [a, b]| a, b \in K\rangle$.

I have a theorem i feel like i should be able to use but i'm not sure how...this is it
Let $\displaystyle H\leq G$ and let $\displaystyle x, y\in G$ then $\displaystyle H\trianglelefteq G \iff [H,G]=\leq H$

and

if $\displaystyle G'\leq H$ then $\displaystyle H\trianglelefteq G$.

any ideas?

2. Originally Posted by delilahjam
here's the prob

Prove if K is a normal subgroup of a group G then $\displaystyle K'\trianglelefteq G$ where $\displaystyle K'=[K,K] =\langle [a, b]| a, b \in K\rangle$.

I have a theorem i feel like i should be able to use but i'm not sure how...this is it
Let $\displaystyle H\leq G$ and let $\displaystyle x, y\in G$ then $\displaystyle H\trianglelefteq G \iff [H,G]=\leq H$

and

if $\displaystyle G'\leq H$ then $\displaystyle H\trianglelefteq G$.

any ideas?
I think that $\displaystyle g[a,b]g^{-1} = [gag^{-1},gbg^{-1}]$.

3. Originally Posted by ThePerfectHacker
I think that $\displaystyle g[a,b]g^{-1} = [gag^{-1},gbg^{-1}]$.
I agree, but I can't just take a single commutator. The elements of K' are of the form $\displaystyle [a_1,b_1]^{r_1}[a_2,b_2]^{r_2}\cdots[a_n,b_n]^{r_n}$ for some $\displaystyle r_1, r_2\cdots \in \mathbb{Z}$

4. Originally Posted by delilahjam
I agree, but I can't just take a single commutator. The elements of K' are of the form $\displaystyle [a_1,b_1]^{r_1}[a_2,b_2]^{r_2}\cdots[a_n,b_n]^{r_n}$ for some $\displaystyle r_1, r_2\cdots \in \mathbb{Z}$
it's a simple observation that if a subgroup $\displaystyle H$ of a group $\displaystyle G$ is generated by a set of elements, say $\displaystyle X,$ then $\displaystyle H$ is normal in $\displaystyle G$ if and only if $\displaystyle gxg^{-1} \in H,$ for all $\displaystyle g \in G, \ x \in X.$

the reason is that if $\displaystyle h \in H,$ then $\displaystyle h=x_1^{\pm 1} x_2^{\pm 1} \cdots x_n^{\pm 1},$ for some $\displaystyle x_i \in X.$ but then: $\displaystyle ghg^{-1}=(gx_1g^{-1})^{\pm 1}(gx_2 g^{-1})^{\pm 1} \cdots (gx_ng^{-1})^{\pm 1}.$ i think you see my point now!

5. awesome thanks!

6. actually thanks from me!

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