# Questions tagged [nilpotent-groups]

A nilpotent group is: 1) a group, whose upper central series stabilizes after a finite length at the whole group. 2) a group, whose lower central series stabilizes after a finite length at the trivial subgroup. 3) a group, that possesses a central series. All those definitions are equivalent. To be used with the tag [group-theory].

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Suppose that all non-normal abelian subgroups of a finite group $G$ are cyclic. What can I say about non-normal nilpotent subgroups of $G$? Is it true that such supgroups are cyclic? I appreciate ...
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### A $p$-group of exponent $p$

I saw a theorem For odd $p$, a $p$-group possesses a characteristic subgroup $D$ of class at most $2$ and of exponent $p$ such that every nontrivial $p’$-automorphism of $G$ induces a nontrivial ...
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### can you help me about my algebra lesson [duplicate]

let n be a square free positive integer. prove that Zn has no nonzero nilpotent elements.
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### Schur Zassenhaus theorem for nilpotent groups

I know that Schur Zassenhaus theorem is valid for any finite group, but my professor said that if a group is nilpotent group, the proof of Schur Zassenhaus theorem becomes very easy. However, I couldn'...
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### Index of $G^n$ in $G$, an $s$-step nilpotent group of rank $\leq r$

For a group $G$ and $n \in \mathbb N$, let $G^n = \langle g^n \mid g \in G \rangle$ I am asked to show that if $G$ is $s$-step nilpotent and of rank at most $r$, then $[G:G^n] \leq n^{O_{r,s}(1)}$ ...
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### If $G$ is $s$-step nilpotent and $n \in \mathbb N$, then $(G_s)^{n^s} \subset (G^n)_s \subset (G_s)^n$

Define the following "commutators" recursively: $[g,h] = g^{-1}h^{-1}gh$ $[g_1, \dots, g_m] = [g_1, [g_2, \dots, g_m]]$ for all $m \geq 3$ Let $G_i$ be the lower central series of a group, $G$. ...
Let $F^{(r)}$ be the free group generated by $r$ elements. Let $\gamma_n(F^{(r)})$ denote its lower central series. Finally, let $F_{n,r} = F^{(r)}/\gamma_{n+1}(F^{(r)})$ be the free nilpotent group ...