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

The Correct Option is B

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**.