Abstract Algebra I · Quotient groups and direct products
Kernels under a composite surjection
Problem
Let $G\xrightarrow{f}H\xrightarrow{g}K$ be surjective homomorphisms. Prove $\ker f\le\ker(g\circ f)$ and show $G/\ker(g\circ f)\cong K$.
Hint
If $f(x)=e_H$, then $g(f(x))=e_K$.
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