query: $G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.
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: 62454
negative_passages.text: Let $f : G → G_1$ be a surjective homomorphism (also called epimorphism) from $G$ to another group $G_1$. Prove that $f(Z(G)) \subseteq Z(G_1)$.
negative_passages.docid: 5297
negative_passages.text: If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
negative_passages.docid: 88928
negative_passages.text: $F:G\to B$ is an isomorphism between groups implies $F^{-1}$ is an isomorphism
negative_passages.docid: 25439
negative_passages.text: f(a) = inverse of a is an isomorphism iff a group G is Abelian
negative_passages.docid: 27728
negative_passages.text: Question on a homomorphism of a set G.
negative_passages.docid: 13761
negative_passages.text: Automorphisms in $\mathbb{Z}_n$
negative_passages.docid: 158396
negative_passages.text: Finite $G$ with involutory automorphism $\alpha$ with no nontrivial fixed points. Proving properties of $\alpha$.
negative_passages.docid: 67554
negative_passages.text: In a group, the number of non-self-inverse elements is even
negative_passages.docid: 24181
negative_passages.text: Showing that $G$ is a group under an alternative operation.
negative_passages.docid: 64269
negative_passages.text: Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
negative_passages.docid: 111168
negative_passages.text: Automorphisms of a finite non-abelian group
negative_passages.docid: 84132
negative_passages.text: Number of preimages of a group element under a homomorphism.
negative_passages.docid: 128443
negative_passages.text: Isomorphism between finite groups
negative_passages.docid: 117468
negative_passages.text: $o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
negative_passages.docid: 114221
negative_passages.text: A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
negative_passages.docid: 129452
negative_passages.text: Groups $G,H$ with surjective homomorphisms that are quotient of each other