query: Let $n$ be a fixed integer and let $ H $ = {$x \in G : x^n = e$}. Prove that H is a subgroup of G.
positive_passages.docid: 154398
positive_passages.text: Show that H is a subgroup of G
negative_passages.docid: 2761
negative_passages.text: If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
negative_passages.docid: 142375
negative_passages.text: Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
negative_passages.docid: 97928
negative_passages.text: Normal subgroup of a normal subgroup
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: 107935
negative_passages.text: If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
negative_passages.docid: 5042
negative_passages.text: Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
negative_passages.docid: 42389
negative_passages.text: If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
negative_passages.docid: 79933
negative_passages.text: Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
negative_passages.docid: 100605
negative_passages.text: let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
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: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
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: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 51145
negative_passages.text: Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?