query: Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and |K| = s. Show that G contains a cyclic group of order lcm(r, s).
positive_passages.docid: 61379
positive_passages.text: Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
negative_passages.docid: 85508
negative_passages.text: Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 51352
negative_passages.text: Subgroups of a finite abelian group
negative_passages.docid: 132475
negative_passages.text: Simple method the show that a group of order 15 is cyclic
negative_passages.docid: 41731
negative_passages.text: Question about the order of the product of two elements in an abelian group
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
negative_passages.docid: 65968
negative_passages.text: Prove that any two cyclic groups of the same order are isomorphic?
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 147574
negative_passages.text: Prove of a finite cyclic group of order $n$ is isomorphic tto $(\mathbb Z_n,+)$.
negative_passages.docid: 65900
negative_passages.text: Show that a finite group with certain automorphism is abelian
negative_passages.docid: 89382
negative_passages.text: In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
negative_passages.docid: 28362
negative_passages.text: Product of two cyclic groups is cyclic iff their orders are co-prime
negative_passages.docid: 143550
negative_passages.text: Order of product of two elements in an abelian group
negative_passages.docid: 83654
negative_passages.text: Order of an element in a finite cyclic group
negative_passages.docid: 4648
negative_passages.text: Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$
negative_passages.docid: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n