Paper 3, Section II, D

Groups | Part IA, 2019

State and prove Lagrange's Theorem.

Hence show that if GG is a finite group and gGg \in G then the order of gg divides the order of GG.

How many elements are there of order 3 in the following groups? Justify your answers.

(a) C3×C9C_{3} \times C_{9}, where CnC_{n} denotes the cyclic group of order nn.

(b) D2nD_{2 n} the dihedral group of order 2n2 n.

(c) S7S_{7} the symmetric group of degree 7 .

(d) A7A_{7} the alternating group of degree 7 .

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