query: Every finite group has a a sequence of nested subgroups where adjacent elements are normal
positive_passages.docid: 142472
positive_passages.text: Every finite group has a composition series
negative_passages.docid: 133770
negative_passages.text: Subgroups of finite abelian groups.
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: 89508
negative_passages.text: Existence of normal subgroups for a group of order $36$
negative_passages.docid: 41237
negative_passages.text: $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
negative_passages.docid: 25404
negative_passages.text: Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
negative_passages.docid: 36040
negative_passages.text: A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
negative_passages.docid: 67272
negative_passages.text: Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
negative_passages.docid: 51352
negative_passages.text: Subgroups of a finite abelian group
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: 23941
negative_passages.text: Normal Subgroups that Intersect Trivially
negative_passages.docid: 114590
negative_passages.text: Prove that any finite group $G$ of even order contains an element of order $2.$
negative_passages.docid: 1360
negative_passages.text: Every group of order $150$ has a normal subgroup of order $25$
negative_passages.docid: 106660
negative_passages.text: Every subgroup of the integers has finite index
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: 92750
negative_passages.text: Groups where all elements are order 3
negative_passages.docid: 64269
negative_passages.text: Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$