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**Statement:**

Let $\phi: G\rightarrow G'$ be a group homomorphism, and let $H=Ker(\phi)$.Let $a\in G$.Then the set

(1)is the left coset $aH$ of $H$,and is also the right coset $Ha$ of $H$.Consequently,the two partitions of $G$ into left cosets and into right cosets of $H$ are the same.

**Proof:**

We want to show that

(There is a standard way to show that two sets are equal;show that each is a subset of the other.)

Suppose that $\phi(x)=\phi(a)$.Then

where $e'$ is the identity of $G'$. By Theorem 13.12, we know that $\phi(a)^{-1}=\phi(a^{-1})$,so we have

(4)Since $\phi$ is a homomorphism,we have

(5)so $\phi(a^{-1}x)=e'$.

But this shows that $a^{-1}x$ is in $H=Ker(\phi)$,so $a^{-1}x=h$ for some $h\in H$,and $x=ah\in aH$.This shows that

To show containment in the other direction,let $y\in aH$,so that $y=ah$ for some $h\in H$.Then

(7)so that $y\in\{x\in G|\phi(x)=\phi(a)\}$.