# Answer to Question #21099 in Abstract Algebra for Akash

Question #21099

If a group G has only three elements, show that it must be abelian.

Expert's answer

As G is a group (with elements a,b,c), there must beidentity element among a b and c. Let it be a. That means

ab=ba=b

ac=ca=c

For other elements there must be inverse to them insidegroup. It easy to understand that b and c are inverse each to other

as a is identity element we have

b^(-1) = c

bc=cb=a

Thus, we just show that group G is abelian as for any 2elements g1, g2 we have

g1g2=g2g1

ab=ba=b

ac=ca=c

For other elements there must be inverse to them insidegroup. It easy to understand that b and c are inverse each to other

as a is identity element we have

b^(-1) = c

bc=cb=a

Thus, we just show that group G is abelian as for any 2elements g1, g2 we have

g1g2=g2g1

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