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.

• You don't need to figure out what a general element in the commutator subgroup looks like. 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). Commented May 6, 2012 at 18:52
• A "direct" argument: If $c$ is in the commutator subgroup, and $g \in G$, then $gcg^{-1}c^{-1}$ is a commutator, hence, by closure, $(gcg^{-1}c^{-1})c = gcg^{-1}$ is in the commutator subgroup. I saw this somewhere in Mathematics Magazine many years ago.
– Ted
Commented May 6, 2012 at 18:57
• Ted's argument is from L. Myers, "Normality of the Commutator Subgroup" Math. Mag. 68 (1995), p. 49. The same slick method shows any subgroup containing the commutator subgroup is a normal subgroup.
– KCd
Commented May 6, 2012 at 19:55
• @harlekin Every normal subgroup of $G$ is the kernel of a homomorphism $f$ from $G$ to some other group $H$. What happens to the commutator subgroup under $f$ if you take $H$ an abelian group? Commented May 6, 2012 at 21:17

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.

If $$c \in G'$$ and $$g \in G$$, then also $$[g, c] = gc g^{-1} c^{-1} \in G'$$. Because G' is closed under products, we would also have $$(gc g^{-1} c^{-1})c \in G'$$. But $$(gc g^{-1} c^{-1})c = gcg^{-1} \in G'$$, so by definition $$G'$$ is a normal subgroup of $$G$$.

• It's not enough characters to make an edit but you forgot dollar signs around the second G' on the first line. Commented Jul 15, 2021 at 1:31

To prove that $$G^{c}$$ is a normal subgroup of $$G$$, it suffices to prove that if $$(g,[c,d])\in{G}\times{G^{c}}$$, then $$g[c,d]g^{-1}\in{G^{c}}$$. But, $$g[c,d]g^{-1} = [gcg^{-1}, gdg^{-1}].$$ Since $$G^{c}$$ is comprised by finite products of commutators in $$G$$, for any $$x\in{G^{c}}$$, it must be the case that $$gxg^{-1}\in{G^{c}}$$ for any $$g\in{G}.$$

if $${ghg^{-1} \in H}$$ for all $${g \in G}$$ and for all $${h \in H}$$ , implies $${gHg^{-1} =H}$$, then H is a normal subgroup.

• we only have to prove that $${g(xyx^{-1}y^{-1})g^{-1} \in H'}$$ for all in G and x,y in G. since $${gxyx^{-1}y^{-1}g^{-1}= ((gx)y(gx)^{-1}y^{-1}) \rightarrow (ygy^{-1}g^{-1}) \in H' }$$, we are done

Take $$x,y,g\in G$$. Then,
$$\begin{equation*} gxyx^{-1}y^{-1}g^{-1}=gxg^{-1}(gy)x^{-1}(gy)^{-1}=(gxg^{-1}x^{-1})(x(gy)x^{-1}(gy)^{-1}), \end{equation*}$$ which is clearly in the commutator of $$G$$. Hence, the commutator subgroup of $$G$$ is normal.

Let $$f:G \rightarrow G$$ be any automorphism. $$f(ghg^{-1}h^{-1})=f(g)f(h)f(g)^{-1}f(h)^{-1}$$. Thus, $$f(G^c) \subset G^c$$ for every automorphism $$f$$. In particular, this holds for every inner automorphism. Thus, $$G^c \triangleleft G$$