(a) (b) If G/Z(G) is cyclic, then G is abelian. Using the statement, prove that the...

   

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(a) (b) If G/Z(G) is cyclic, then G is abelian. Using the statement, prove that the center Z(G) of a non-abelian group G of order 33 is {e}. Let Ss be the symmetric group of permutations of five numbers and D60 be the dihedral group of symmetries of regular 60-sided polygon. (i) (ii) Show that the two groups S5 and D60 have the same number of elements. Prove that for every element S5, the order of is less than or equal to 6. (iii) Show that S5 is not isomorphic to D60. (a) (b) If G/Z(G) is cyclic, then G is abelian. Using the statement, prove that the center Z(G) of a non-abelian group G of order 33 is {e}. Let Ss be the symmetric group of permutations of five numbers and D60 be the dihedral group of symmetries of regular 60-sided polygon. (i) (ii) Show that the two groups S5 and D60 have the same number of elements. Prove that for every element S5, the order of is less than or equal to 6. (iii) Show that S5 is not isomorphic to D60.