query: Show that the map $f(a)=a^2$ is an isomorphism of groups from G to itself.
positive_passages.docid: 75473
positive_passages.text: If a group $G$ has odd order, then the square function is injective.
negative_passages.docid: 38153
negative_passages.text: Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
negative_passages.docid: 97608
negative_passages.text: Showing a Ring of endomorphisms is isomorphic to a Ring
negative_passages.docid: 4228
negative_passages.text: Prove that the only automorphism of $\mathbb{Q}(\sqrt[3]{2})$ is the identity automorphism.
negative_passages.docid: 106091
negative_passages.text: Showing $\mathcal{P}(X)$ is isomorphic to $2^X$
negative_passages.docid: 46863
negative_passages.text: Show that $g$ is injective
negative_passages.docid: 130746
negative_passages.text: The symbol for a map which is an isomorphism?
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 7890
negative_passages.text: Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
negative_passages.docid: 26687
negative_passages.text: Show that $f$ maps the unit circle to itself
negative_passages.docid: 33196
negative_passages.text: show that A(T) is a group under operation of composition of functions
negative_passages.docid: 158975
negative_passages.text: Isomorphism of centers of isomorphic groups
negative_passages.docid: 67234
negative_passages.text: The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
negative_passages.docid: 125582
negative_passages.text: Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
negative_passages.docid: 52556
negative_passages.text: Isomorphism of $S^1$.
negative_passages.docid: 94571
negative_passages.text: How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
negative_passages.docid: 126930
negative_passages.text: Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$