query: Normal extension of a group
positive_passages.docid: 157496
positive_passages.text: Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$
negative_passages.docid: 131204
negative_passages.text: A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
negative_passages.docid: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup
negative_passages.docid: 89508
negative_passages.text: Existence of normal subgroups for a group of order $36$
negative_passages.docid: 22444
negative_passages.text: Normal subgroups of dihedral groups
negative_passages.docid: 80186
negative_passages.text: How to show that $K\leq S_4$ is a normal subgroup?
negative_passages.docid: 54911
negative_passages.text: The preimage of a normal subgroup under a group homomorphism is normal
negative_passages.docid: 125244
negative_passages.text: Equivalent definitions of normal subgroups
negative_passages.docid: 57227
negative_passages.text: Maximal normal $\pi$-subgroups and torsion subgroups
negative_passages.docid: 103423
negative_passages.text: Normal subgroup where G has order prime
negative_passages.docid: 157819
negative_passages.text: Every group of order $56$ has a proper non - trivial normal subgroup?
negative_passages.docid: 58205
negative_passages.text: $ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
negative_passages.docid: 7973
negative_passages.text: Groups of order $pq$ have a proper normal subgroup
negative_passages.docid: 20386
negative_passages.text: Any subgroup of index $p$ in a $p$-group is normal.
negative_passages.docid: 68436
negative_passages.text: Groups in derived series are normal in the whole group
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: 18182
negative_passages.text: An abelian group $G$ with ${\rm Aut}(G)$ non-abelian