If G is a group that (ab)n = an.bn , for all a, b ϵG and for three consecutive integers n. Then prove that G is an abelian group.
Proof:
Let a and b be any two elements of G.
Suppose n, n+1 , n + e are three consecutive integers such that
(ab)n = an.bn …………….(1)
(ab)n+1 = an+1.bn+1 ………..(2)
(ab)n+2 = an+2. bn+2……………(3)
Equation (2) can also be written as,
(ab)(ab)n = a.an.b.bn
a.b.an.bn = a.an.b.bn
b.an = an.b [by cancellation law]
Equation (3) can also be written as,
(ab)(ab)n+1 = a.an+1.b.bn+1
a.b.an+1.bn+1 = a.an+1.b.bn+1
b.an+1 = an+1.b [by cancellation law]
b.an.a = an.a.b
an.b.a = an.a.b
⇒ ba = ab [by left cancellation law]
Therefore, G is an abelian group.
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