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Find all subgroups of a cyclic group of order 12. Also what is the number of generators?

Doubt: I got subgroups as a, a2, a3, a4, a6 and a12. How to identify the identity element among these?

1 Answer

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Cyclic Group Of order = 12 with generator = a has elements {a,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12}

Order of generator = Order of group = 12

Order of an element is the smallest positive integer such that an is identity element.

Identity element = a12

Other Generators are = {a5,a7,a11}

Order of every subgroup divides the order of group.

Order of Subgroup SubGroup
1 a12
2 a6,a12
3 a4,a8,a12
4 a3,a6,a9,a12
6 a2,a4,a6,a8,a10,a12
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