query: Can $gHg^{-1} \subseteq H$ but $gHg^{-1} \neq H$
positive_passages.docid: 41085
positive_passages.text: Conjugate subgroup strictly contained in the initial subgroup?
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 80996
negative_passages.text: A canonical construction proving $\left|HK\right| = \left|H\right|\left|K\right| / \left|H \cap K\right|$?
negative_passages.docid: 105951
negative_passages.text: Does $G\times K\cong H\times K$ imply $G\cong H$?
negative_passages.docid: 144732
negative_passages.text: If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
negative_passages.docid: 105910
negative_passages.text: How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
negative_passages.docid: 132342
negative_passages.text: Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
negative_passages.docid: 62077
negative_passages.text: Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
negative_passages.docid: 110545
negative_passages.text: Space $X$ such that $H_i(X,\mathbb{Z})\neq 0$ but $H_i(X,\mathbb{Z}_3)= 0$
negative_passages.docid: 100987
negative_passages.text: Let f be an r-cycle in $S_n$. Given any $h \in S_n$, show that $hfh^{-1} = (h(x_1), h(x_2), \dots, h(x_r))$
negative_passages.docid: 140962
negative_passages.text: Counterexamples for $G \times H$ isomorphic to $G$ implies $H$ trivial.
negative_passages.docid: 142739
negative_passages.text: Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
negative_passages.docid: 70278
negative_passages.text: $A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ -- how to prove this?
negative_passages.docid: 122581
negative_passages.text: Prove or disprove that $G_1/H_1 \cong G_2/H_2$
negative_passages.docid: 81116
negative_passages.text: $gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
negative_passages.docid: 19396
negative_passages.text: For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?
negative_passages.docid: 9854
negative_passages.text: If $U$ is a neighborhood of $e$, there exists $V$ neighborhood of $e$ such that $V^{-1}V \subseteq U$