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Group Theory : Symmetry Groups, Cyclic Subgroups and Permutations

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Thisquestion is concerned with subgroups ofthe group S5 of permutations on the set {1,2,3,4,5} , a group with 120 elements.
(a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6, and give examples of each of these.
Explain why this group does not have cyclic subgroups of any other order.
(8 marks
(b) By considering the symmetry groups of appropriate geometric figures, give examples of:
(1) a subgroup of order 4 that is not cyclic;
(ii) a subgroup of order 6 that is not cyclic;
(iii) a subgroup oforder 8.
(6 marks)
(c) By considering those permutations that fix one element, or otherwise, give an example ofa subgroup oforder 24 and another oforder 12. [You need not list all the elements ofthese groups, but you should explain clearly which elements constitute each subgroup.]
(6 marks)
(d) List the potential orders of subgroups of S5 (other than S5 itself), according to Lagrange's theorem, in addition to those already considered in this question. Give an example of a subgroup of one of these orders.
(5 marks)

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Symmetry Groups, Cyclic Subgroups and Permutations are investigated. The solution is detailed and well presented.

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Problem #1
(a) We know, each element in can be expressed as disjoint cycles. The order of the element is the least common multiple of the lengths of those cycles. So we have the following ...

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