Question

Let a, b be elements of the group G. Assume that A has order 5 and a3b = ba3, then G is

• A. Both abelian and cyclic group
• B. Non-abelian group
• C. Cyclic group
• D. Abelian group

Answer 1

Answer: (B)

Non-abelian group

Answer 2

Given:
1. The order of $( a )$ is 5. This means $( a^5 = e )$ (identity element) and $( a^n \neq e )$ for $( n < 5 ).$
2$. ( a^3b = ba^3 ).$
From the given condition $( a^3b = ba^3 ),$ it is evident that $( ab = ba ),$ which implies that the operation is commutative for these elements.
However, just because two elements commute doesn't mean that all elements of the group $( G )$ commute with each other. So, we can't conclude that the group is abelian based on these two elements alone.
The given information doesn't provide any evidence that $( G )$ is cyclic either. Hence, the only conclusion we can make from the provided information is that these specific elements ( a) and ( b ) commute, but it does not guarantee anything about the nature of the entire group $( G ).$ None of the provided options $( A ), ( C ),$ and $( D )$ can be conclusively determined based on the given information. The closest possible answer would be: B) Non-abelian group**.