query: Abstract algebra-normal subgroups
positive_passages.docid: 19352
positive_passages.text: Normal subgroup of index m contains every element of order n where m and n are relatively prime
negative_passages.docid: 44165
negative_passages.text: the image of normal subgroups
negative_passages.docid: 7973
negative_passages.text: Groups of order $pq$ have a proper normal subgroup
negative_passages.docid: 125244
negative_passages.text: Equivalent definitions of normal subgroups
negative_passages.docid: 15630
negative_passages.text: Normal subgroup of prime order is characteristic
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?
negative_passages.docid: 16804
negative_passages.text: Prove whether or not a subgroup is normal in the dihedral group $D_4$
negative_passages.docid: 133298
negative_passages.text: Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.
negative_passages.docid: 13523
negative_passages.text: Are normal subgroups transitive?
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: 4883
negative_passages.text: Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
negative_passages.docid: 92837
negative_passages.text: Solvable and nilpotent groups, normal series and intuition
negative_passages.docid: 115204
negative_passages.text: Show that SO(n) is a normal subgroup of O(n)
negative_passages.docid: 49527
negative_passages.text: Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
negative_passages.docid: 157819
negative_passages.text: Every group of order $56$ has a proper non - trivial normal subgroup?
negative_passages.docid: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup
negative_passages.docid: 12589
negative_passages.text: Show that every subgroup of the quaternion group Q is a normal subgroup of Q