In Group Theory from Abstract Algebra, if G is a group and a ∈ G has order n (|a| = n), then the cyclic subgroup generated by a ∈ G, written＜a＞= {a^n | n ∈ ℤ}, has distinct elements {e, a, a^2, a^3, ..., a^(n-1)}. Moreover, a^i = a^j if and only if n divides i - j. How is this proved? Use the Division Algorithm! We also stuy part of the proof that ＜a^k＞ = ＜a^gcd(n,k)＞ and |a^k| = n/gcd(n,k). In particular, a^k is a generator of ＜a＞, where |a| = n, iff n and k are relatively prime. • Abstract Algebra Cours... .
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5 сен 2024
