 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-2/8/classification/solution

If an element has order six, it generates

G

, so

G ≅ C_{6}

. Otherwise choose an element

x

of order three. The preceding result excludes any further order-three elements outside

{e, x, x^{2}}

, and there are no elements of order six by assumption. Each of the other three elements must therefore have order two. Thus

G ≅ C_{6}, or G = {e, x, x^{2}, h_{1}, h_{2}, h_{3}}, ord (x) = 3, ord (h_{i}) = 2.

The noncyclic case is the symmetric group on three points, as the classification of a group of order six further identifies.

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