query: Is it true that $\operatorname{Gal}(K/K^G)=G$ even when $G$ is infinite?
positive_passages.docid: 155935
positive_passages.text: Examples where $H\ne \mathrm{Aut}(E/E^H)$
negative_passages.docid: 101079
negative_passages.text: Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
negative_passages.docid: 67637
negative_passages.text: How come $\lambda(\mathbb{Q})= \inf\left\{\sum_{n \ge 1} \lambda([a_n,b_n)) : \mathbb{Q} \subset \bigcup_{n \ge 1} [a_n,b_n)\right\} = 0$?
negative_passages.docid: 45469
negative_passages.text: If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
negative_passages.docid: 110217
negative_passages.text: $GL(n,K)$ is open in $M(n,K)$
negative_passages.docid: 98816
negative_passages.text: How to prove that $\{f+g<a\}=\bigcup_{r\in{\Bbb Q}}(\{f<r\}\cap\{g<a-r\})$?
negative_passages.docid: 140471
negative_passages.text: Is $ G \cong G/N \times N$?
negative_passages.docid: 91287
negative_passages.text: Is $G/pG$ is a $p$-group?
negative_passages.docid: 55976
negative_passages.text: Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
negative_passages.docid: 40383
negative_passages.text: Is it true that $|g| = |\phi(g)|$ for all homomorphisms $\phi: G \to G$ and $g \in G$?
negative_passages.docid: 123847
negative_passages.text: Cardinality of $GL_n(K)$ when $K$ is finite
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: 57846
negative_passages.text: Multiplicative inverse in $\operatorname{GF}(2^8)$?
negative_passages.docid: 22758
negative_passages.text: If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.
negative_passages.docid: 120401
negative_passages.text: Does there exist a pair of infinite fields, the additive group of one isomorphic to the multiplicative group of the other?
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?