Paper 3, Section II, D

Groups | Part IA, 2013

(a) Let GG be the dihedral group of order 4n4 n, the symmetry group of a regular polygon with 2n2 n sides.

Determine all elements of order 2 in GG. For each element of order 2 , determine its conjugacy class and the smallest normal subgroup containing it.

(b) Let GG be a finite group.

(i) Prove that if HH and KK are subgroups of GG, then KHK \cup H is a subgroup if and only if HKH \subseteq K or KHK \subseteq H.

(ii) Let HH be a proper subgroup of GG, and write G\HG \backslash H for the elements of GG not in HH. Let KK be the subgroup of GG generated by G\HG \backslash H.

Show that K=GK=G.

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