Prove that the relation~ R of Theorem 10.1 is an equivalence relation. 

Data from Theorem 10.1

Let H be a subgroup of G. Let the relation ~L be defined on G by a ~ _L b if and only if a^-1 b ∈ H Let ~ _R be defined by a ~ _R b if and only if  ab^-1 ∈ H

Then ~ _L and ~ _R are both equivalence relations on G.

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Proof We show that is an equivalence relation, and leave the proof for ~R to Exercise 26. When reading the proof, notice how we must constantly make use of the fact that H is a subgroup of G. Reflexive Symmetric Transitive Let a € G. Then a ¹a = e ande EH since H is a subgroup. Thus a~La. Suppose ab. Then a ¹b e H. Since H is a subgroup, (a-¹b)-¹ is in H and (a¹b)-¹ = b¹a, so b-la is in H and b~ a. ~ Let a b and b~ c. Then a ¹b e H and b¹c € H. Since H is a subgroup, (a-¹b)(b¹c) = a¹c is in H, so a ~L c.