o(a)=o(a^k).gcd of n and k | some theorems on order of an element | Bsc/Msc Mathematics | Ravina

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Let G be group, a ɛ G and O(a)=n. Then for any positive integer k, O(a^k) =n/(n,k). Let G be group, a ɛ G and O(a)=n. Then for any positive integer k, O(a^k) =n/(n,k).
{a^k}={a^gcd(n,k)} and |a^k|=n/gcd(n,k) {a^k}={a^gcd(n,k)} and |a^k|=n/gcd(n,k)
In a group G, a is an element of order n, then for any natural number k, a^k=e iff n divides k. In a group G, a is an element of order n, then for any natural number k, a^k=e iff n divides k.
[a^k] = [a^gcd(n,k)] and |a^k| = n/gcd(n,k) Proof (Abstract Algebra) [a^k] = [a^gcd(n,k)] and |a^k| = n/gcd(n,k) Proof (Abstract Algebra)
Order of an element || O(a)|O(G) || a^O(G) = e Order of an element || O(a)|O(G) || a^O(G) = e
#Order of element #important theorem: O(ab) = O(a).O(b) if gcd(O(a), O(b)) = 1 and ab=ba #Order of element #important theorem: O(ab) = O(a).O(b) if gcd(O(a), O(b)) = 1 and ab=ba
O(a^k)=O(a)/g.c.d{O(a),k)} O(a^k)=O(a)/g.c.d{O(a),k)}
6. Order of an element of a group || Examples || Group Theory #orderofelements 6. Order of an element of a group || Examples || Group Theory #orderofelements