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Home / Modern Abstract Algebra / Show that if every element of the group G execept the identity element is of order 2, then G is abelian.

Show that if every element of the group G execept the identity element is of order 2, then G is abelian.

 

Solution: Let a, b∈G such that a ≠ e, b ≠ e.

Then according to the question,

a2 = e, b2 = e.

Also ab∈G and So (ab)2 = e

Now (ab)2 = e

⟹ (ab) (ab) = e

⟹ a(ab ab) b = a e b

⟹ a2bab2 = ab

⟹ e ba e = ab

⟹ ba = ab .

Hence G is abelian. (Proved)

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