4. Lagrange's theorem Prove Lagrange's theorem. For each a ∈ G, define the set

$$

aH := {a o h | h ∈ H}.

$$

Show the following.

(a) The set G is the union of all sets aH, where a ranges over G.

(b) For

a, b ∈ G, if the sets aH and bH have a nonempty intersection then they are equal.

(c) For

a, b ∈ G, the sets aH and bH are of the same size. (Hint: Consider the function f: aH → bH with f(a * h) = b * h and argue that f has an inverse.)

(d) Use (a)-

(c) to prove Lagrange's theorem.

(e) What is the equivalence relation R that corresponds to this partition? That is, can you define when aH = bH holds based on an equation in the group in terms of a and b?