query: Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
positive_passages.docid: 97469
positive_passages.text: How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
negative_passages.docid: 112157
negative_passages.text: If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
negative_passages.docid: 56607
negative_passages.text: gcd and order of elements of group
negative_passages.docid: 124438
negative_passages.text: Find $|(\mathbb{Z} \oplus \mathbb{Z}) / \langle(3,6)\rangle|$ and determine if its cyclic
negative_passages.docid: 157806
negative_passages.text: Order of a element on a Symmetric Group
negative_passages.docid: 110390
negative_passages.text: Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
negative_passages.docid: 5069
negative_passages.text: $GL_n(\mathbb F_q)$ has an element of order $q^n-1$
negative_passages.docid: 16476
negative_passages.text: Order of the product of two commuting elements with coprime orders in a group.
negative_passages.docid: 74924
negative_passages.text: identity element of $(D_n ,\gcd)$
negative_passages.docid: 13734
negative_passages.text: Show that every group of prime order is cyclic
negative_passages.docid: 102990
negative_passages.text: Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$
negative_passages.docid: 119758
negative_passages.text: On the order of elements of $\mathrm{GL}(2,q)$.
negative_passages.docid: 53357
negative_passages.text: Cyclic Group Generators of Order $n$
negative_passages.docid: 17178
negative_passages.text: Order of $\phi(g)$ divides the order of $g$
negative_passages.docid: 30676
negative_passages.text: $G'$ and the product of all the elements in an odd order group
negative_passages.docid: 89215
negative_passages.text: How to describe the quotient group $\mathbb{Z} \times \mathbb{Z} / \langle (1, 2) \rangle$