query: Let $H$ be a subgroup of $G$ and $gH$ is in $Hg$ for some $g$. Can we get that $gH=Hg$?
positive_passages.docid: 111024
positive_passages.text: Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?
negative_passages.docid: 11825
negative_passages.text: Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
negative_passages.docid: 85129
negative_passages.text: If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
negative_passages.docid: 34567
negative_passages.text: If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
negative_passages.docid: 128834
negative_passages.text: Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
negative_passages.docid: 77361
negative_passages.text: How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 154878
negative_passages.text: Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
negative_passages.docid: 44942
negative_passages.text: A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
negative_passages.docid: 39469
negative_passages.text: If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
negative_passages.docid: 61379
negative_passages.text: Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
negative_passages.docid: 3756
negative_passages.text: If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
negative_passages.docid: 1220
negative_passages.text: Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.
negative_passages.docid: 105951
negative_passages.text: Does $G\times K\cong H\times K$ imply $G\cong H$?
negative_passages.docid: 103002
negative_passages.text: Group equals union of two subgroups
negative_passages.docid: 145511
negative_passages.text: Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
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 |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?