query: How to prove for a finite group that $| \langle g \rangle|=o(g)$?
positive_passages.docid: 150199
positive_passages.text: How to prove for a finite group that $|\langle g \rangle|=o(g)$?
negative_passages.docid: 137931
negative_passages.text: Problem from "The Theory of Finite Groups" by Kurzweil and Stellmacher
negative_passages.docid: 14890
negative_passages.text: Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
negative_passages.docid: 82935
negative_passages.text: Prove that f∈Θ(g)if and only ifO(f) =O(g).
negative_passages.docid: 27086
negative_passages.text: proof for $\langle q| p \rangle = e^{ipq}$
negative_passages.docid: 132725
negative_passages.text: Is $\mathbb{Z}/\langle n\rangle$ cyclic and of order $n$?
negative_passages.docid: 23517
negative_passages.text: Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group
negative_passages.docid: 108330
negative_passages.text: Product of all elements in finite abelian group equals its own inverse
negative_passages.docid: 59561
negative_passages.text: Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 89215
negative_passages.text: How to describe the quotient group $\mathbb{Z} \times \mathbb{Z} / \langle (1, 2) \rangle$
negative_passages.docid: 2626
negative_passages.text: Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$
negative_passages.docid: 102990
negative_passages.text: Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$
negative_passages.docid: 142421
negative_passages.text: Conjugacy Classes in Finite Groups
negative_passages.docid: 85011
negative_passages.text: Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
negative_passages.docid: 20679
negative_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
negative_passages.docid: 52121
negative_passages.text: Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.