Algebra and Geometry | Part IA, 2001

(i) Let A4A_{4} denote the alternating group of even permutations of four symbols. Let XX be the 3-cycle (123)(123) and P,QP, Q be the pairs of transpositions (12)(34)(12)(34) and (13)(24)(13)(24). Find X3,P2,Q2,X1PX,X1QXX^{3}, P^{2}, Q^{2}, X^{-1} P X, X^{-1} Q X, and show that A4A_{4} is generated by X,PX, P and QQ.

(ii) Let GG and HH be groups and let

G×H={(g,h):gG,hH}G \times H=\{(g, h): g \in G, h \in H\}

Show how to make G×HG \times H into a group in such a way that G×HG \times H contains subgroups isomorphic to GG and HH.

If DnD_{n} is the dihedral group of order nn and C2C_{2} is the cyclic group of order 2 , show that D12D_{12} is isomorphic to D6×C2D_{6} \times C_{2}. Is the group D12D_{12} isomorphic to A4A_{4} ?

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