Can someone give me a hint on how to solve the following question from the textbook Abstract Algebra: Theory and Applications by Thomas W. Judson?
Do I just need to show that $\bar{\phi}$ is a homomorphism using properties from $\phi$?
2026-03-26 01:03:21.1774487001
Show that $\phi$ induces a canonical homomorphism.
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You have a commutative diagram $$ \require{AMScd}\begin{CD}G_1 @>\phi>>G_2\\ @VVV@VVV\\ G_1/H_1@>\bar\phi>>G_2/H_2 \end{CD} $$, where the downward arrows are the canonical projections onto the quotients.
So $\bar\phi(gH_1)=\phi(g)H_2$.