Paper 3 , Section II, E

Groups | Part IA, 2017

State Lagrange's theorem. Show that the order of an element xx in a finite group GG is finite and divides the order of GG.

State Cauchy's theorem.

List all groups of order 8 up to isomorphism. Carefully justify that the groups on your list are pairwise non-isomorphic and that any group of order 8 is isomorphic to one on your list. [You may use without proof the Direct Product Theorem and the description of standard groups in terms of generators satisfying certain relations.]

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