BIGtheme.net http://bigtheme.net/ecommerce/opencart OpenCart Templates
Friday , July 28 2017
Home / Modern Abstract Algebra / show that G is abelian.

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 = ai+1 bi+1

⟹(aibi) (ab) = ai+1 bi+1       [by(i)]

⟹ ai(bia)b = ai(abi)b

⟹ bia = abi ——————————-(iv) [by cancellation law]

Similarly from (iii) we have,

(ab)i+2 = ai+2 bi+2

⟹ (ai+1bi+1) (ab) = ai+2 bi+2

⟹ ai+1(bi+1a)b = ai+1(abi+1)b

⟹ bi+1a = abi+1

⟹ bi+1a = (abi)b = (bia)b [by (iv)]

⟹ bi(ba) = bi (ab)

⟹ ba = ab [by cancellation law]

⟹ G is abelian.                                                      (Proved)

Check Also

Lecture – 4| Abstract Algebra

Problem – 1 : Show that if every element of the group G except the ...

Leave a Reply

Your email address will not be published. Required fields are marked *