Problem – 1 : Show that if every element of the group G except the identity element is of order 2, then G is abelian. Solution: Let a, b ∈ G such that a ≠ e, b ≠ e Then a2 = e, b2 = e. Also ab ∈ G ...

Read More »## 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 ...

Read More »## show that G is abelian.

If G is a group in which (ab)1 = a1 b1 for three consecutive integers I for all a, b ∈G, show that G is abelian. Solution: We have (ab)1 = a1 b1 ——————————-(i) (ab)i+1 = ai+1 bi+1 ————————-(ii) (ab)i+2 = ai+2 bi+2 ————————-(iii) From (ii), we have (ab)i+1 = ...

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