query: existence of $a \neq e$ in group such that $a^2=e$
positive_passages.docid: 114590
positive_passages.text: Prove that any finite group $G$ of even order contains an element of order $2.$
negative_passages.docid: 80849
negative_passages.text: There does not exist group $G$ such that Aut$(G)\cong \mathbb{Z}_n$ (for odd $n$)
negative_passages.docid: 76943
negative_passages.text: Existence of subgroup of order six in $A_4$
negative_passages.docid: 1364
negative_passages.text: Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
negative_passages.docid: 36887
negative_passages.text: Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
negative_passages.docid: 108310
negative_passages.text: $(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
negative_passages.docid: 135701
negative_passages.text: An example of a group such that $G \cong G \times G$
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: 33593
negative_passages.text: An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
negative_passages.docid: 56650
negative_passages.text: Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
negative_passages.docid: 12455
negative_passages.text: $|G|>2$ implies $G$ has non trivial automorphism
negative_passages.docid: 2104
negative_passages.text: Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
negative_passages.docid: 23946
negative_passages.text: Prove the definitions of $e$ to be equivalent
negative_passages.docid: 150199
negative_passages.text: How to prove for a finite group that $|\langle g \rangle|=o(g)$?
negative_passages.docid: 133762
negative_passages.text: A set which satisfies all conditions for a Group except associativity
negative_passages.docid: 61550
negative_passages.text: Existence of elements of order $p^k$ in a finite group
negative_passages.docid: 58654
negative_passages.text: If $|A| > \frac{|G|}{2} $ then $AA = G $