query: $H$ prime order, normal subgroup of group $G$. Prove $H$ in center $Z(G)$.
positive_passages.docid: 19458
positive_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: 111886
negative_passages.text: Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
negative_passages.docid: 33300
negative_passages.text: Showing order of $Z (G) =1$ or $pq$
negative_passages.docid: 105139
negative_passages.text: Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center
negative_passages.docid: 70373
negative_passages.text: Intersection of normal subgroups proof
negative_passages.docid: 65037
negative_passages.text: $H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
negative_passages.docid: 99205
negative_passages.text: Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
negative_passages.docid: 53475
negative_passages.text: Let $H$ have order $m$ and $K$ have order $n$, where $m$ and $n$ are relatively prime. Then $H \cap K=\{e\}$
negative_passages.docid: 19720
negative_passages.text: A normal subgroup intersects the center of the $p$-group nontrivially
negative_passages.docid: 85011
negative_passages.text: Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
negative_passages.docid: 19352
negative_passages.text: Normal subgroup of index m contains every element of order n where m and n are relatively prime
negative_passages.docid: 124309
negative_passages.text: $g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
negative_passages.docid: 134353
negative_passages.text: A group of order $108$ has a proper normal subgroup of order $\geq 6$.
negative_passages.docid: 38853
negative_passages.text: Cyclic group of prime order
negative_passages.docid: 95534
negative_passages.text: If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
negative_passages.docid: 122970
negative_passages.text: A question on $p$-groups, and order of its commutator subgroup.
negative_passages.docid: 86336
negative_passages.text: For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists