Discussion
Which of the following statements is not true ?
Answer: C
The false statement is Every group of order 12 is cyclic.
The alternating group \(A_{4}\), the dihedral group of order 12 and \(\mathbf{Z}_{2} \times \mathbf{Z}_{6}\) all have order 12 and none is cyclic; \(A_{4}\) is not even abelian.
The other statements are true. There is a field with 4 elements, \(\mathbf{F}_{4} = \mathbf{Z}_{2}[x]/(x^{2} + x + 1)\), since a finite field exists for every prime power. A Boolean ring with 8 elements is \(\mathbf{Z}_{2} \times \mathbf{Z}_{2} \times \mathbf{Z}_{2}\), or the power set of a three-element set under symmetric difference and intersection. Every group of order 15 = 3 × 5 is cyclic, hence abelian, because 3 does not divide 5 - 1.
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