Algebra and Geometry | Part IA, 2003

(i) State the orbit-stabilizer theorem for a group GG acting on a set XX.

(ii) Denote the group of all symmetries of the cube by GG. Using the orbit-stabilizer theorem, show that GG has 48 elements.

Does GG have any non-trivial normal subgroups?

Let LL denote the line between two diagonally opposite vertices of the cube, and let

H={gGgL=L}H=\{g \in G \mid g L=L\}

be the subgroup of symmetries that preserve the line. Show that HH is isomorphic to the direct product S3×C2S_{3} \times C_{2}, where S3S_{3} is the symmetric group on 3 letters and C2C_{2} is the cyclic group of order 2 .

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