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College Algebra Exam Review 111

College Algebra Exam Review 111 - 2.5 COSETS AND LAGRANGES...

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2.5. COSETS AND LAGRANGE’S THEOREM 121 This computation requires a little labor. If you want, you can get a computer to do some of the repetitive work; for example, programs for computations in the symmetric group are distributed with the symbolic mathematics program Mathematica . With the notation H D f 2 S 4 W .4/ D 4 g , the results are eH D .1 2/H D .1 3/H D .2 3/H D .1 2 3/H D .1 3 2/H D H .4 3/H D .4 3 2/H D .2 1/.4 3/H D .2 4 3 1/H D .4 3 2 1/H D .4 3 1/H D f .4 3/; .4 3 2/; .2 1/.4 3/; .2 4 3 1/; .4 3 2 1/; .4 3 1/ g .4 2/H D .3 4 2/H D .4 2 1/H D .4 2 3 1/H D .3 4 2 1/H D .3 1/.4 2/H D f .4 2/; .3 4 2/; .4 2 1/; .4 2 3 1/; .3 4 2 1/; .3 1/.4 2/ g .4 1/H D .4 1/ .3 2/H D .2 4 1/H D .2 3 4 1/H D .3 2 4 1/H D .3 4 1/H D f .4 1/; .4 1/ .3 2/; .2 4 1/; .2 3 4 1/; .3 2 4 1/; .3 4 1/ g : The regularity of the preceding data for left cosets of subgroups of symmetric groups is striking! Based on these data, can you make any
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