Math 4124Monday, February 28February 28, Ungraded HomeworkExercise 3.5.11 on page 111Prove thatS4has no subgroup isomorphic toQ8.InQ8we have the relationi j=k, and of coursei,j,kall have order 4; in other words we havethe product of two elements of order 4 giving an element of order 4. Therefore ifQ8wasisomorphic to a subgroup ofS4, we would have inS4the product of two elements of order4 giving an element of order 4. Since an element of order 4 inS4must be a 4-cycle, whichis an odd permutation, we would now have the product of two odd permutations giving anodd permutation. But the product of two odd permutations is always an even permutation,so we have a contradiction. ThereforeQ8cannot be isomorphic to a subgroup ofS4.LetG=D8, the dihedral group of order 8. Prove thatGhas exactly one normal subgroup oforder 2, and that there are exactly 5 normal subgroups containing this subgroup.
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