Left and right cosets of kernel of group homomorphism

Question

On page 76 of Abstract Algebra by Dummit and Foote, we can read in Proposition 2:

Let $\phi: G\to H$ be a homomorphism of groups with kernel $K$. Let $X\in G/K$ be the fiber above $a$, i.e., $X=\phi^{-1}(a)$. Then

$\ \ \ $ (1) For any $u \in X$, $X=\{uk:k\in K\}$

$\ \ \ $ (2) For any $u \in X$, $X=\{ku:k\in K\}$

What I'm wondering about is that if we take some element not in $X$ but still in $G$, will the right cosets of the kernel still be equal to the left cosets of the kernel?

Answer 1

Yes. The kernel of a group homomorphism is a normal subgroup, and for a normal subgroup the left and right cosets are the same. (In fact, a subgroup is normal if and only if its left and right cosets are the same.)