How to show that the commutator subgroup is a normal subgroup

It is suggested as an exercise in Serge Lang’s book “Algebra” to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.

I’d like to do that but I am afraid I need help,

I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.

That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.

But here I get stuck, first because I am unsure how to write a general element in $G^c$ – a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this – I am sure there is one, but somehow I am blind today.

The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?

An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ – here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.

Thanks for your hints !!

Answer

Denote the commutator of $a$ and $b$ by $a^{-1}b^{-1}ab = [a,b]$.

If $u$ is an element from the commutator subgroup, then $g^{-1}ug = u(u^{-1}g^{-1}ug) = u[u, g]$ .


Another approach: the commutator subgroup is defined to be the subgroup generated by the commutators, so every element of the commutator subgroup is of the form $$[a_1, b_1][a_2,b_2]\ldots[a_n, b_n].$$ It is enough to show that $g^{-1}[a,b]g$ is always in the commutator subgroup, because then

$$g^{-1}[a_1, b_1][a_2,b_2]\ldots[a_n, b_n]g = (g^{-1}[a_1, b_1]g)(g^{-1}[a_2,b_2]g)(g^{-1}\ldots g)(g^{-1}[a_n, b_n]g)$$

is a product of elements from the commutator subgroup. When $\phi$ is any homomorphism, we have $\phi([a,b]) = [\phi(a), \phi(b)]$. Since for any $g \in G$ the map $\phi$ defined by $\phi(x) = g^{-1}xg$ is a homomorphism, the result follows.

Attribution
Source : Link , Question Author : harlekin , Answer Author : Mikko Korhonen

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