query: If $G/N$ is nilpotent and $N\leq Z(G)$, then $G$ is nilpotent
positive_passages.docid: 45310
positive_passages.text: If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
negative_passages.docid: 5069
negative_passages.text: $GL_n(\mathbb F_q)$ has an element of order $q^n-1$
negative_passages.docid: 34183
negative_passages.text: An equivalent condition with $\{0\}$ being the only nilpotent ideal
negative_passages.docid: 42389
negative_passages.text: If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
negative_passages.docid: 156443
negative_passages.text: if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
negative_passages.docid: 49768
negative_passages.text: If Gal(K,Q) is abelian then |Gal(K,Q)|=n
negative_passages.docid: 14403
negative_passages.text: If $n = p_1^{a_1} ... p_r^{a_r}$ the set of nilpotents of $\mathbb{Z}_n$ is $(p_1 p_2 ... p_{r-1} p_r)\Bbb Z_n$
negative_passages.docid: 30637
negative_passages.text: Exponent of $GL(n,q)$.
negative_passages.docid: 61868
negative_passages.text: Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.
negative_passages.docid: 129610
negative_passages.text: Why is the direct product of a finite number of nilpotent groups nilpotent?
negative_passages.docid: 127603
negative_passages.text: If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$
negative_passages.docid: 80601
negative_passages.text: Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
negative_passages.docid: 15387
negative_passages.text: Order of $GL(n, \mathbb Z_p)$
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: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
negative_passages.docid: 33945
negative_passages.text: Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
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) $?