Question

Which of the following is true for all non-abelian groups?

• A. \(O(G)>7\)
• B. \(O(G)\geq 9\)
• C. \(O(G)>6\)
• D. \(O(G)\geq 6\)

Hint:

The smallest non-abelian group is \(S_3\), whose order is \(6\).

Answer 1

The Correct Option is D

Concept:
A group is called abelian if \[ ab=ba \] for all elements \(a,b\in G\). A group is non-abelian if at least two elements do not commute. The smallest order of a non-abelian group is \(6\).

Step 1: Recall groups of small order. All groups of order \[ 1,2,3,4,5 \] are abelian. For example, every group of prime order is cyclic, and every cyclic group is abelian.

Step 2: Smallest non-abelian example. The symmetric group \[ S_3 \] has order \[ 3!=6 \] and it is non-abelian. For example, permutations do not always commute.

Step 3: General conclusion. Since the smallest possible order of a non-abelian group is \(6\), every non-abelian group must satisfy \[ O(G)\geq 6 \]

Step 4: Final answer. \[ \boxed{O(G)\geq 6} \]