query: Proof: Well defined multiplication in groups
positive_passages.docid: 18308
positive_passages.text: Coset multiplication giving a well defined binary operation
negative_passages.docid: 145123
negative_passages.text: Prove a group generated by two involutions is dihedral
negative_passages.docid: 8742
negative_passages.text: Examples and further results about the order of the product of two elements in a group
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 24181
negative_passages.text: Showing that $G$ is a group under an alternative operation.
negative_passages.docid: 98507
negative_passages.text: $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$
negative_passages.docid: 36117
negative_passages.text: Prove that matrices of the form $\begin{pmatrix} x & x \\ x & x \end{pmatrix}$ are a group under matrix multiplication.
negative_passages.docid: 29802
negative_passages.text: Part of simple proof of nontrivial center in p-group
negative_passages.docid: 46674
negative_passages.text: How can I prove that every group of $N = 255$ elements is commutative?
negative_passages.docid: 3462
negative_passages.text: Proving a set of automorphisms of a group is a group under composition
negative_passages.docid: 39425
negative_passages.text: Surprising but simple group theory result on conjugacy classes
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: 64527
negative_passages.text: Show that an operation of elements in a finite group contains each element exactly once
negative_passages.docid: 5741
negative_passages.text: Direct product of abelian groups
negative_passages.docid: 17997
negative_passages.text: Right identity and Right inverse implies a group
negative_passages.docid: 121077
negative_passages.text: Product of all elements in finite group
negative_passages.docid: 18160
negative_passages.text: How to show that the group of nonzero real numbers under multiplication is not isomorphic to the group of real numbers under addition?