query: What is the group $G=\langle a,b\mid a^2=b^2=1\rangle$?
positive_passages.docid: 88718
positive_passages.text: A group is generated by two elements of order $2$ is infinite and non-abelian
negative_passages.docid: 130792
negative_passages.text: Commutator of a group
negative_passages.docid: 117815
negative_passages.text: Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.
negative_passages.docid: 122665
negative_passages.text: Help finding the fundamental group of $S^2 \cup \{xyz=0\}$
negative_passages.docid: 90437
negative_passages.text: Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
negative_passages.docid: 82037
negative_passages.text: Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
negative_passages.docid: 67772
negative_passages.text: How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$
negative_passages.docid: 73113
negative_passages.text: Groups and Symmetries
negative_passages.docid: 36142
negative_passages.text: What is $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$ isomorphic to?
negative_passages.docid: 57166
negative_passages.text: Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question
negative_passages.docid: 33025
negative_passages.text: What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
negative_passages.docid: 60140
negative_passages.text: Proof of Equivalence Relation $R=\{(a,b)\in\Bbb N^2: 5\mid (2a+3b)\}$
negative_passages.docid: 19683
negative_passages.text: Is a semigroup $G$ with left identity and right inverses a group?
negative_passages.docid: 13631
negative_passages.text: Showing group with $p^2$ elements is Abelian
negative_passages.docid: 120495
negative_passages.text: Does this correctly interpret the "$\cong$" between groups as "$=$" between sets?
negative_passages.docid: 150585
negative_passages.text: Group with $a=a^{-1}$ for all $a\in G$ is abelian
negative_passages.docid: 139218
negative_passages.text: Is this group solvable?