query: If $G/Z(G)$ is nilpotent then $G$ is nilpotent
positive_passages.docid: 25748
positive_passages.text: If $G / Z(G)$ nilpotent then G is nilpotent.
negative_passages.docid: 112701
negative_passages.text: Prove that if matrix $A$ is nilpotent, then $I+A$ is invertible.
negative_passages.docid: 43580
negative_passages.text: Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 125299
negative_passages.text: If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.
negative_passages.docid: 133019
negative_passages.text: Nilpotent elements in $\mathbb Z_n$
negative_passages.docid: 43645
negative_passages.text: Show that $G/H\cong S_3$
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: 145254
negative_passages.text: $G$ be a non-nilpotent and supersoluble finite group.
negative_passages.docid: 90811
negative_passages.text: If $[G' : G'']\leq p^2$, then $G'$ is abelian.
negative_passages.docid: 59561
negative_passages.text: Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
negative_passages.docid: 25009
negative_passages.text: $\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
negative_passages.docid: 76317
negative_passages.text: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
negative_passages.docid: 98739
negative_passages.text: If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
negative_passages.docid: 28460
negative_passages.text: Show matrix is nilpotent
negative_passages.docid: 39788
negative_passages.text: $a+1,a-1$ invertible for nilpotent element