MATH 436 Notes: Cyclic groups and Invariant Subgroups
نویسنده
چکیده
Thus if x has infinite order then the elements in the set {x|k ∈ Z} are all distinct while if x has order o(x) ≥ 1 then x = e exactly when k is a multiple of o(x). Thus x = x whenever m ≡ n modulo o(x). By the First Isomorphism Theorem, < x >∼= Z/o(x)Z. Hence when x has finite order, | < x > | = o(x) and so in the case of a finite group, this order must divide the order of the group G by Lagrange’s Theorem.
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