Paper 3, Section II, D

Groups | Part IA, 2014

Let SnS_{n} be the group of permutations of {1,,n}\{1, \ldots, n\}, and suppose nn is even, n4n \geqslant 4.

Let g=(12)Sng=(12) \in S_{n}, and h=(12)(34)(n1n)Snh=(12)(34) \ldots(n-1 n) \in S_{n}.

(i) Compute the centraliser of gg, and the orders of the centraliser of gg and of the centraliser of hh.

(ii) Now let n=6n=6. Let GG be the group of all symmetries of the cube, and XX the set of faces of the cube. Show that the action of GG on XX makes GG isomorphic to the centraliser of hh in S6S_{6}. [Hint: Show that 1G-1 \in G permutes the faces of the cube according to hh.]

Show that GG is also isomorphic to the centraliser of gg in S6S_{6}.

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