query: Normal subgroups that have only the identity in common commute with each other.
positive_passages.docid: 130455
positive_passages.text: If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
negative_passages.docid: 145511
negative_passages.text: Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?
negative_passages.docid: 129830
negative_passages.text: A group with no proper non-trivial subgroups
negative_passages.docid: 55350
negative_passages.text: H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
negative_passages.docid: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup
negative_passages.docid: 48607
negative_passages.text: If $A$ is normal and $A$ and $B$ commute, then $A^*$ and $B$ commute
negative_passages.docid: 42582
negative_passages.text: Infinite group with only two conjugacy classes
negative_passages.docid: 15630
negative_passages.text: Normal subgroup of prime order is characteristic
negative_passages.docid: 139096
negative_passages.text: $|AxA|$ is a constant implies normal subgroup
negative_passages.docid: 91874
negative_passages.text: Existence of finite indexed normal subgroup for a given finite indexed subgroup.
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: 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: 63780
negative_passages.text: to show characteristic subgroup is normal in G
negative_passages.docid: 5024
negative_passages.text: Why do we define quotient groups for normal subgroups only?
negative_passages.docid: 51359
negative_passages.text: Prove that every subgroup of an abelian group is a normal subgroup.
negative_passages.docid: 7640
negative_passages.text: Exactly one nontrivial proper subgroup