Algebra and Geometry | Part IA, 2004

Let G=g,hh2=1,g6=1,hgh1=g1G=\left\langle g, h \mid h^{2}=1, g^{6}=1, h g h^{-1}=g^{-1}\right\rangle be the dihedral group of order 12 .

i) List all the subgroups of GG of order 2 . Which of them are normal?

ii) Now list all the remaining proper subgroups of GG. [There are 6+36+3 of them.]

iii) For each proper normal subgroup NN of GG, describe the quotient group G/NG / N.

iv) Show that GG is not isomorphic to the alternating group A4A_{4}.

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