query: Prove that $G$ has a element whose order is least common multiple of $m$ and $n$.
positive_passages.docid: 9393
positive_passages.text: Order of elements is lcm-closed in abelian groups
negative_passages.docid: 15633
negative_passages.text: If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
negative_passages.docid: 28749
negative_passages.text: Proving that any common multiple of two numbers is a multiple of their least common multiple
negative_passages.docid: 26449
negative_passages.text: $|G|=24$ prove that $G$ is not simple
negative_passages.docid: 49294
negative_passages.text: prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
negative_passages.docid: 6410
negative_passages.text: Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
negative_passages.docid: 100061
negative_passages.text: The Maximum possible order for an element $S_n$
negative_passages.docid: 81864
negative_passages.text: Finite group of order $mn$ with $\gcd(m,n) = 1$ .
negative_passages.docid: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 142718
negative_passages.text: How can I find the smallest set of groups of $n$ elements such that every element is in the same group as every other at least once?
negative_passages.docid: 77064
negative_passages.text: Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
negative_passages.docid: 5089
negative_passages.text: An element of a group has the same order as its inverse
negative_passages.docid: 104513
negative_passages.text: Order of elements in $Z_n$
negative_passages.docid: 16476
negative_passages.text: Order of the product of two commuting elements with coprime orders in a group.
negative_passages.docid: 114590
negative_passages.text: Prove that any finite group $G$ of even order contains an element of order $2.$
negative_passages.docid: 2731
negative_passages.text: Groups of even order have an element whose order is 2.
negative_passages.docid: 53475
negative_passages.text: Let $H$ have order $m$ and $K$ have order $n$, where $m$ and $n$ are relatively prime. Then $H \cap K=\{e\}$