query: Prove that if $A,B$ are finite Abelian groups and for every $n$ they have the same amount of elements of order $n$, then $A \simeq B$
positive_passages.docid: 59811
positive_passages.text: Finite Abelian groups with the same number of elements for all orders are isomorphic
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 119808
negative_passages.text: Proving that $n$-cycle has order $n$
negative_passages.docid: 74586
negative_passages.text: Prove that every group of order $4$ is abelian
negative_passages.docid: 51359
negative_passages.text: Prove that every subgroup of an abelian group is a normal subgroup.
negative_passages.docid: 145159
negative_passages.text: Finite group is abelian if the representatives of its conjugacy classes commute
negative_passages.docid: 81354
negative_passages.text: The product of all the elements of a finite abelian group
negative_passages.docid: 145127
negative_passages.text: For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
negative_passages.docid: 70394
negative_passages.text: Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
negative_passages.docid: 65900
negative_passages.text: Show that a finite group with certain automorphism is abelian
negative_passages.docid: 13355
negative_passages.text: Product of elements of a finite abelian group
negative_passages.docid: 100605
negative_passages.text: let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
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.
negative_passages.docid: 15526
negative_passages.text: Question on groups of order $pq$
negative_passages.docid: 110708
negative_passages.text: On proving that a finitely generated group has a finite number of subgroups with index $n$.
negative_passages.docid: 152662
negative_passages.text: Proving that any finite cyclic group of order $n$ is isomorphic to $$
negative_passages.docid: 114590
negative_passages.text: Prove that any finite group $G$ of even order contains an element of order $2.$