query: A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
positive_passages.docid: 19720
positive_passages.text: A normal subgroup intersects the center of the $p$-group nontrivially
negative_passages.docid: 122266
negative_passages.text: equivalent definition of normal subgroup
negative_passages.docid: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
negative_passages.docid: 89508
negative_passages.text: Existence of normal subgroups for a group of order $36$
negative_passages.docid: 113936
negative_passages.text: Showing that a finite abelian group has a subgroup of order m for each divisor m of n
negative_passages.docid: 147253
negative_passages.text: Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
negative_passages.docid: 142894
negative_passages.text: Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
negative_passages.docid: 20386
negative_passages.text: Any subgroup of index $p$ in a $p$-group is normal.
negative_passages.docid: 157819
negative_passages.text: Every group of order $56$ has a proper non - trivial normal subgroup?
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: 134353
negative_passages.text: A group of order $108$ has a proper normal subgroup of order $\geq 6$.
negative_passages.docid: 139611
negative_passages.text: Can a normal subgroup of a finite nonabelian group be nonabelian?
negative_passages.docid: 58694
negative_passages.text: Which non-abelian finite groups have the property that every subgroup is normal?
negative_passages.docid: 7108
negative_passages.text: Normal subgroup of prime index
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: 101735
negative_passages.text: A finite $p$-group cannot be simple unless it has order $p$
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?