Prove that the relation~ R of Theorem 10.1 is an equivalence relation. Data from Theorem 10.1 Let
Question:
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.
Transcribed Image Text:
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.
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