# 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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### Some calculation in $F_5[D_{30}]$.

I am currently reading a research paper and have encountered a point that I am struggling to understand. In the paper, it is proven that $J(F_5[D_{30}])^5 = (0)$, indicating that the Jacobson radical ...
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### Commutator of two elements in group algebra $\mathbb F_{5}D_{30}.$

I want to understand how to find the commutator of two elements in the group algebra $\mathbb{F}_{5}D_{30}$ using GAP. Additionally, I would like to determine the nilpotency class of the nilpotent ...
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### Non-Abelian group of exponent $3$ and Nilpotent class $2$ .

Let $G$ be a non-Abelian group such that $G^3 = 1$ and $G$ is a nilpotent class $2$ group, with order $3^{32}$. Our task is to determine the structure of the group $G$ or identify any information ...
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### If $G$ is nilpotent of class $n$, then $G/Z(G)$ is nilpotent of class $n−1$.

I saw this here: If $G$ is nilpotent of class $c$, then $G/Z(G)$ is nilpotent of class $c-1$. However, the definition of Nilpotent I have to work with is equivalent but different. A group is Nilpotent ...
1 vote
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### Classification of torsion-free nilpotent groups of class 2

Some background Let $G$ be a torsion-free nilpotent group of class $2$ and rank $2$ (i.e., generated by two elements). Then, $G$ has to be isomorphic to the Heisenberg group. This is relatively easy ...
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### What is the index of $H$ in $G=\langle H,g\rangle$, if $G$ is nilpotent and $g$ has finite order?

Suppose we have a nilpotent group $G$ that is generated by a subgroup $H$ and a single element $g$ of finite order. Is $H$ necessarily of finite index in $G$? What can be said of this index in any ...
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### Class of nilpotency of semidirect product

Let $G=N\rtimes H$ for some groups $N$ and $H$. Is it true that the nilpotency class of $G$ is at most maximum of nilpotency classes of $N$ and $H$, if it is known that both groups are nilpotent? And ...
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### Example of a quasi nilpotent element which is not a nilpotent element

Let $R$ be a ring with unity. An element $a\in R$ is said to be a quasi nilpotent element of $R$ if $1-ax$ is unit for all $x\in comm(a)$ where $comm(a)=\lbrace x \in R | ax=xa\rbrace$. It is obvious ...
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### Nilpotent quotients of a residually nilpotent group

I am working on a problem that requires taking quotients by normal subgroups that do not intersect a finite set of members of a group $G$. Is there a known collection of groups which always allows ...
1 vote
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### Fitting length of Dihedral groups

Let $G$ be a dihedral group of order $2n$ where $n\geq 1$, denoted by $D_n$. We know that $G$ is nilpotent if and only if $n=2^i$ for all $i\geq 1$, a proof of this you can check in the below link 1. ...
1 vote
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### The $i^{th}$ term in the upper central series of the dihedral group of order $2^n$ is equal to the $(n-1-i)^{th}$ term in its lower central series.

The Problem: Prove that $Z_i(D_{2^n})=D_{2^n}^{n-1-i}$, where $D_{2^n}$ is the dihedral group of order $2^n$. $Z_i(G)$ is the $i^{th}$ term in the upper central series of $G$, which is inductively ...
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### How do subgroups of the inner automorphisms of a group look like?

I'm trying to prove the following proposition: Let $G$ be a group. Then $G$ is nilpotent iff ${\rm Inn}(G)$ is nilpotent. I've proven that if $G$ is nilpotent then ${\rm Inn}(G)$ is nilpotent as ...
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### Self-centralizing subgroups of order $p^2$ in $p$-groups of maximal class

Let $G$ be a finite $p$-group. It's an old result of Suzuki that if $G$ possesses a self-centralizing subgroup of order $p^2$ then $G$ has maximal class, and in fact the converse is true. However, the ...
1 vote
### Proof Verification: $G/Z(G)$ is nilpotent implies $G$ is nilpotent
I wrote a proof down for the fact that if $G/Z (G)$ is nilpotent then $G$ is nilpotent and I wonder if this is correct. I would appreciate it if someone verifies this and/or suggests some alternative ...