query: If $H \le G$ and $H$ has index 2 in $G$ then $a^2 \in H, \forall a\in G$
positive_passages.docid: 144732
positive_passages.text: If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
negative_passages.docid: 27728
negative_passages.text: Question on a homomorphism of a set G.
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 150042
negative_passages.text: Prove $x^{-1}Hx$ is a subgroup of $G$
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: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 58863
negative_passages.text: Proof that $A_n$ the only subgroup of $ S_n$ index $2$.
negative_passages.docid: 102979
negative_passages.text: If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
negative_passages.docid: 146760
negative_passages.text: Prove that H is a subgroup of G
negative_passages.docid: 34902
negative_passages.text: If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
negative_passages.docid: 91774
negative_passages.text: Existence of group $G$ with $|G|>2$ and automorphism group of odd order
negative_passages.docid: 75568
negative_passages.text: Subgroup of index $2$ is Normal
negative_passages.docid: 143095
negative_passages.text: If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
negative_passages.docid: 98145
negative_passages.text: If $G$ is cyclic then $G/H$ is cyclic?
negative_passages.docid: 53409
negative_passages.text: Show group of order $4n + 2$ has a subgroup of index 2.
negative_passages.docid: 65037
negative_passages.text: $H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.