query: A normal subgroup of a nilpotent group intersects the centre non-trivially
positive_passages.docid: 153921
positive_passages.text: Infinite nilpotent group, any normal subgroup intersects the center nontrivially
negative_passages.docid: 112263
negative_passages.text: an example for $\pi$-quasinormal subgroup but is not normal
negative_passages.docid: 90449
negative_passages.text: Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
negative_passages.docid: 33728
negative_passages.text: Basis for Center of Group Ring
negative_passages.docid: 15630
negative_passages.text: Normal subgroup of prime order is characteristic
negative_passages.docid: 139451
negative_passages.text: Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection
negative_passages.docid: 33945
negative_passages.text: Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
negative_passages.docid: 92837
negative_passages.text: Solvable and nilpotent groups, normal series and intuition
negative_passages.docid: 49257
negative_passages.text: How to show that Klein four-group is a normal subgroup of the alternating group $A_4$
negative_passages.docid: 45028
negative_passages.text: How to show that $\langle a \rangle$ is a subgroup of a group $G$, where $a$ is in center of the group $G$
negative_passages.docid: 104509
negative_passages.text: The center of $A_n$ is trivial for $n \geq 4$
negative_passages.docid: 36040
negative_passages.text: A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
negative_passages.docid: 6134
negative_passages.text: Some Subgroup of Dihedral Group is Normal
negative_passages.docid: 42891
negative_passages.text: Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
negative_passages.docid: 7640
negative_passages.text: Exactly one nontrivial proper subgroup
negative_passages.docid: 141265
negative_passages.text: Finding a normal and not normal subgroup of $S_3$
negative_passages.docid: 27215
negative_passages.text: Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$