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Let G be a group with the following property Whenever a, b, and c b.pdf

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This document proves that if a group G has the property that whenever elements a, b, c satisfy ab = ca, then b = c, it follows that G is an abelian (commutative) group. It does this by assuming elements a, b, c satisfy the cancellation property ab = ca, then showing this implies b = c and b commutes with a, so the group is abelian.

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Let G be a group with the following property: Whenever a, b, and c belong to G and ab = ca, then b = c. Prove that G is Abelian. ("Cross" cancellation implies commutativity.) Solution Let (G, *) be a group. If for any a, b in G we have a*b = b*a, the group is abelian (or commutative). You have a,b and c in G, the operation is (.) You have defined a.b = c.a Let's find b.a b.a = a.c, a.c = c.a Then, (G, .) is abelian.

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