The Right And Left Cosets Of The Kernel Are Equal

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Problem 83 (The Right And Left Cosets Of The Kernel Are Equal)

Suppose that $f:G\to H$ is a homomorphism. Let $K$ be the kernel of $f$ and let $a\in G$. Prove the following:

1. The right coset $Ka$ equals the set of values $b\in G$ such that $f(b)=f(a)$, or symbolically we have $$Ka = \{b\in G\mid f(b)=f(a)\} = f^{-1}(f(a)).$$
2. The left coset $aK$ equals the same set $\{b\in G\mid f(b)=f(a)\}$, which means the left and right cosets are the same or $Ka=aK$.

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