Discussion
Let G be a non-abelian group with n elements, then which of the following values of n is possible :
Answer: C
The possible value is \(n = 10\).
The dihedral group \(D_{5}\), the symmetries of a regular pentagon, has 10 elements and is non-abelian: a rotation and a reflection do not commute.
Each of the other orders forces every group to be abelian. A group of prime order 13 is cyclic. A group of order \(9 = 3^{2}\) has order \(p^{2}\), and every group of order \(p^{2}\) is abelian. A group of order \(15 = 3 \times 5\) is cyclic, because 3 does not divide \(5 - 1 = 4\), so the Sylow theorems leave only one Sylow subgroup of each order, both normal.
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