Let (A,+) and (B,+) be commutative groups and suppose that A is isomorphic to B. Prove that $A/dA$ is isomorphic to $B/dB$, where $d \in \mathbb{N}$.
Any thoughts on this one?
Let (A,+) and (B,+) be commutative groups and suppose that A is isomorphic to B. Prove that $A/dA$ is isomorphic to $B/dB$, where $d \in \mathbb{N}$.
Any thoughts on this one?
Let $\phi: G\to H$ is our isomorphism. Define $\phi|_{dG}:dG\to H$ and show that it is an injective homomorphism and so is between $dG$ and $dH$. Moreover, $\phi|_{dG}$ is onto (Why?). For your problem, regarding to @Marc's answer, define $\psi:G\to \frac{H}{dH}$ with $\psi(g)=\phi(g)+dH$ and use the first isomorphism theorem for it.
Morphisms of commutative groups commute with multiplication by an integer. So if $\phi:A\to B$ is an isomorphism then $\phi(dA)=dB$, and so $\phi$ and its inverse pass to the quotient, giving isomorphisms there.