query: Proof Help: In a group $G$, there exists a $g$ such that $g^2 = e$
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: 55976
negative_passages.text: Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
negative_passages.docid: 98474
negative_passages.text: Show the existence of a certain subgroup of F2
negative_passages.docid: 79452
negative_passages.text: How to prove that it is a group?
negative_passages.docid: 12455
negative_passages.text: $|G|>2$ implies $G$ has non trivial automorphism
negative_passages.docid: 102859
negative_passages.text: Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.
negative_passages.docid: 135701
negative_passages.text: An example of a group such that $G \cong G \times G$
negative_passages.docid: 140334
negative_passages.text: Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
negative_passages.docid: 31252
negative_passages.text: "Efficient version" of Cayley's Theorem in Group Theory
negative_passages.docid: 10403
negative_passages.text: Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 14890
negative_passages.text: Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
negative_passages.docid: 90811
negative_passages.text: If $[G' : G'']\leq p^2$, then $G'$ is abelian.
negative_passages.docid: 99630
negative_passages.text: For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
negative_passages.docid: 30676
negative_passages.text: $G'$ and the product of all the elements in an odd order group
negative_passages.docid: 108830
negative_passages.text: Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
negative_passages.docid: 127603
negative_passages.text: If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$