Question

Any group of order 3 is
1. cyclic and abelian
2. cyclic but not abelian
3. infinite cyclic group
4. none of these

Answer 1

Correct Answer - Option 1 : cyclic and abelian

Concept:

• Any group of order 3 is cyclic.
• Or Any group of three elements is an abelian group.
• The group has 3 elements: 1, a, and b. ab can’t be a or b, because then we’d have b=1 or a=1. So ab must be 1. The same argument shows ba=1. So ab=ba, and since that’s the only nontrivial case, the group is also abelian.

• Every group of prime order is cyclic.
• If an abelian group of order 6 contains an element of order 3, then it must be a cyclic group.
• Every subgroup of a cyclic group is itself a cyclic group.
• Every proper subgroup of an infinite cyclic group is infinite.