(March 2015) Let \(G_1,G_2\) be groups, let \(H_2\subseteq G_2\) be a subgroup, and let \(\varphi:G_1\to G_2\) be a homomorphism. Define \[ H_1=\{x\in G_1\mid \varphi(x)\in H_2\}. \] It is a fact, which you may assume, that \(H_1\) is a subgroup of \(G_1\). Prove that for any \(x,y\in G_1\), \(H_1x=H_1y\) if and only if \(H_2\varphi(x)=H_2\varphi(y)\).
Solution
Proof. (=⇒) Given x,y ∈G1 such that H1x =H1y, we have xy−1∈H1 [by the right coset relation for H1]. So φ(x)φ(y)−1 =φ ( xy−1) ∈H2, by definition of H1. Thus, [by the right coset relation for H2], we have H2φ(x) =H2φ(y). (⇐=) Given x,y ∈ G1 such that H2φ(x) = H2φ(y), we have φ(x)φ(y)−1∈ H2 [by the right coset relation for H2]. Thus, φ ( xy−1) =φ(x)φ(y)−1∈H2, and therefore xy−1∈ H1, by definition of H1. Thus, [by the right coset relation for H1], we have H1x =H1y. QED