query: H is a subgroup of a finite group G, and if |G| = m|H|, adapt the proof of Lagrange's Theorem to show that g^m! ∈ H for all g ∈ 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: 152071
negative_passages.text: Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
negative_passages.docid: 4184
negative_passages.text: Is Lagrange's theorem the most basic result in finite group theory?
negative_passages.docid: 56245
negative_passages.text: Let H be a subgroup of a group G. Prove that the following statements are equivalent.
negative_passages.docid: 140334
negative_passages.text: Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 24966
negative_passages.text: Converse of Lagrange's theorem for abelian groups
negative_passages.docid: 21673
negative_passages.text: If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
negative_passages.docid: 101345
negative_passages.text: Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
negative_passages.docid: 64269
negative_passages.text: Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
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: 51145
negative_passages.text: Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
negative_passages.docid: 106779
negative_passages.text: How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
negative_passages.docid: 19458
negative_passages.text: Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
negative_passages.docid: 22477
negative_passages.text: If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
negative_passages.docid: 18181
negative_passages.text: Complete classification of the groups for which converse of Lagrange's Theorem holds