query: Proving that G is abelian
positive_passages.docid: 59128
positive_passages.text: If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
negative_passages.docid: 3907
negative_passages.text: Abelian quotient group
negative_passages.docid: 11957
negative_passages.text: G is the product of its Sylow subgroups
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G
negative_passages.docid: 26449
negative_passages.text: $|G|=24$ prove that $G$ is not simple
negative_passages.docid: 60208
negative_passages.text: Proof of property of 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: 18415
negative_passages.text: Direct sum of Abelian groups and Isomorphism
negative_passages.docid: 152630
negative_passages.text: showing that a group of order 45 is abelian
negative_passages.docid: 73571
negative_passages.text: Proof of Cauchy's Theorem for abelian groups
negative_passages.docid: 13631
negative_passages.text: Showing group with $p^2$ elements is Abelian
negative_passages.docid: 94239
negative_passages.text: Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
negative_passages.docid: 40287
negative_passages.text: Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian
negative_passages.docid: 81354
negative_passages.text: The product of all the elements of a finite abelian group
negative_passages.docid: 49527
negative_passages.text: Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
negative_passages.docid: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.