Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 3 / 6D / c

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- Solution c

Solution

An element of order three in

S_{7}

3 1^{4}

or

3^{2} 1

. There are

(3 7) (3 - 1)! = 35 \cdot 2 = 70

single three-cycles. For two disjoint three-cycles, choose the two unordered three-element supports and an orientation on each:

\frac{1}{2} (3 7) (3 4) (2) (2) = 280.

Hence

S_{7}

contains

350

elements of order three.

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