Let G be a group and H a normal subgroup of G. Define a relation R...

Transcribed Image Text:

Let G be a group and H a normal subgroup of G. Define a relation R on G x G by x Ry if and only if ry E H. (a) Show that R is an equivalence relation on G. (b) For a, a, y, bE G: (i) Show that if x Ry and aRb, then raRyb. (ii) Show that if a Ry then r-Ry. (c) Prove that [e]R = H, i.e. the equivalence class of e under R is H. In particular, describe [x]R for every x E G. Let G be a group and H a normal subgroup of G. Define a relation R on G x G by x Ry if and only if ry E H. (a) Show that R is an equivalence relation on G. (b) For a, a, y, bE G: (i) Show that if x Ry and aRb, then raRyb. (ii) Show that if a Ry then r-Ry. (c) Prove that [e]R = H, i.e. the equivalence class of e under R is H. In particular, describe [x]R for every x E G.