Answer to Question #21099 in Abstract Algebra for Akash

Question

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

Answer 1

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