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Equivalent definition of equivalence relation on a group

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Show that the following are equivalent:

(a) ~ is an equivalence relation on a group G

(b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a.

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A detailed proof of the equivalence of the standard definition of equivalence relation on a group and the alternative definition given in the statement of this problem is presented in the solution.

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Let G be a group, and let ~ be a (binary) relation on G. We need to show that (a) implies (b), and that (b) implies (a).

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First, we show that (a) implies (b).

So assume (a), namely, that ~ is an equivalence relation on G.

By definition, an equivalence relation is reflexive, so ~ is reflexive. Thus it suffices to show that, for all a, b, c in G: if a ~ b and b ~ c, then c ~ a.

So let a, b, c in G, such that a ~ b and b ~ c. We need to show that c ~ a.

By definition, an equivalence relation is transitive, by which we mean that, for all x, y, z in G: if x ~ y and y ~ ...

Solution provided by:
Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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