Question

In an abelian group G, if the order of an element of
'a' is 4 and order of an element b is 3. Then order of
(ab)^4 is​

Answer 1

SOLUTION

GIVEN

In an abelian group G, if the order of an element of

'a' is 4 and order of an element b is 3.

TO DETERMINE

The order of (ab)⁴

EVALUATION

Let e be the identity element

Here it is given that the order of an element of

'a' is 4 and order of an element b is 3.

$\sf{ {a}^{4} = e}$

$\sf{ {b}^{3} = e }$

Now we have to find the order of (ab)⁴

Now

$\sf{ { \bigg(( {ab)}^{4} \bigg) }^{3} }$

$\sf{ = ( {ab)}^{12} }$

$\sf{ = {a}^{12} \times {b}^{12} }$

$\sf{ = {( {a}^{4})}^{3} \times {( {b}^{3} )}^{4} }$

$\sf{ = {( e)}^{3} \times {(e)}^{4} }$

$\sf{= e}$

Hence the order of (ab)⁴ is 3

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Prove that the inverse of the product of two elements of a group is the product of the inverses taken in the reverse order

https://brainly.in/question/22739109

2. prove that - evey cauchy sequence is bounded.

https://brainly.in/question/26180669