Let i, a, and b be distinct elements of the group
where i is the identity. Let the group operation
be concatenation. Assume that ab ≠ ba.
ab = i or ba = i --> ab = ba ><
ab = a or ba = a --> b = i ><
ab = b or ba = b --> a = i ><
Therefore, i, a, b, ab, and ba must be the
five elements of the group.
a(ba) = a --> ba = i ><
a(ba) = ab --> (ab)a = ab --> a = i ><
a(ba) = ba --> a = i ><
a(ba) = i --> (ab)a = (ba)a --> ab = ba ><
a(ba) = b --> (ab)a = b --> a(ab) ≠ i
--> aa = i --> a(ab) = (aa)b = b
--> a(ab) = a(ba) --> ab = ba ><
Since a(ba) is not an element of the group,
our original assumption that ab ≠ ba must
be false. Thus the group must be commutative.
Note: >< denotes a contradiction.
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