# Commutator Subgroup is Normal Subgroup of Kernel of Homomorphism

Let $G$ be a group, $H$ an abelian group, $\phi : G \rightarrow H$ a homomorphism. Show that $C(G) \lhd \mathrm{Ker}(\phi)$

I must be misunderstanding something, because what if we have the natural (bijective) homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$. Then the kernel of the homomorphism is $0$ but the commutator subgroup of $\mathbb{Z}$ is $\mathbb{Z}$.

Any help would be appreciated.

• What's the commutator subgroup of an abelian group? What are the elements of the form $ghg^{-1}h^{-1}$ in an abelian group? – PVAL-inactive Feb 25 '14 at 23:52

It seems as if you are misunderstanding the definition of the commutator. It is not the center of the group.

Many times denoted $[G,G]$, it is defined to be the subgroup generated by the set $\{xyx^{-1}y^{-1} : x, y \in G\}$, where I am using multiplicative notation.

In your example $\mathbb{Z}$, an element in the commutator will be, for example with $x = 5$ and $y = 7$: $5 + 7 - 5 - 7$. You should be able to answer PVAL's question above, and this should give an indication of how to solve the problem.

• The subgroup generated by such expressions! – Ted Shifrin Feb 26 '14 at 1:07
• That is hardly a subgroup! – Pedro Tamaroff Feb 26 '14 at 2:41
• @TedShifrin Ack! Thank you! – RghtHndSd Feb 26 '14 at 3:09

If $$x\in C(G)$$ then $$x$$ is of the form $$x=aba^{-1}b^{-1}$$ for some $$a,b \in G$$. Now $$\phi (x)= \phi(a)\phi(b)\phi(a)^{-1}\phi(b)^{-1}$$ is in $$H$$, and since $$H$$ is abelian we have $$\phi(x)=\phi(a)\phi(a)^{-1}\phi(b)\phi(b)^{-1}=1$$ and thus $$x \in \ker(\phi)$$.

• You perhaps meant $x\in \ker \phi$. – Diogenes Feb 27 '14 at 17:35
• @Diogenes : of course. thanks! – Ludolila Feb 27 '14 at 18:23

Claim If $x=[a,b]$ then $\varphi(x)=[\varphi(a),\varphi(b)]$.

Proof [insert proof here]

Corollary For every group morphism $\eta :G\to H$ with $H$ abelian, $[G,G]\subseteq \ker \eta$.

• This is related to a theorem that every normal subgroup has the commutator as a subset. In fact, because it is a theorem that N is a normal subgroup iff it is the kernel of a group homomorphism, it follows that one can deduce from your corollary that every normal subgroup must have the commutator as a subset. – kakashi10192020 Nov 19 '16 at 4:06