Let $G$ be a finitely generated nilpotent group. Let $Z = Z(G)$ be the center of the group $G$. Is $Z$ finitely generated?
2026-03-26 07:59:08.1774511948
Is it true that the center of a finitely generated nilpotent group is finitely generated?
231 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in FINITELY-GENERATED
- projective module which is a submodule of a finitely generated free module
- Ascending chain of proper submodules in a module all whose proper submodules are Noetherian
- Is a connected component a group?
- How to realize the character group as a Lie/algebraic/topological group?
- Finitely generated modules over noetherian rings
- Integral Elements form a Ring
- Module over integral domain, "Rank-nullity theorem", Exact Sequence
- Example of a module that is finitely generated, finitely cogenerated and linearly compact, but not Artinian!
- Computing homology groups, algebra confusion
- Ideal Generated by Nilpotent Elements is a Nilpotent Ideal
Related Questions in NILPOTENT-GROUPS
- On the finite non-nilpotent group
- Quotient of finite, nilpotent group by Frattini subgroup is isomorphic to product of quotients of Sylow subgroups by their respective Frattini groups.
- Solvable, non-nilpotent group with nilpotent commutator subgroup
- When is the automorphism group of a finite $p$-group nilpotent?
- Locally compact nilpotent group has an open subgroup isomorphic to $\mathbb{R}^n\times K$
- Let $G$ be a nilpotent group of class 3. Then for every $x ,y ,z$ in $G$, $[x,y,z][y,z,x][z,x,y]=1$.
- Use an alternative definition for nilpotent groups and show that $D_{8}$ is nilpotent.
- If $G/Z(G)$ is nilpotent then $G$ is nilpotent
- Let $Z_i(G)$ be the terms of the upper central series of $G$. Let $H \trianglelefteq G$. Show $Z_i(G)H/H \subseteq Z_i(G/H)$
- Connected Lie group for which every connected Lie subgroup is simply connected
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Yes it is true. A finitely generated nilpotent group is Noetherian, i.e. any subgroup is finitely generated. In particular the center is finitely generated.
To see why a finitely generated nilpotent group is Noetherian, first show that the property of being Noetherian is closed under group extensions. Next, any term in the lower central series is finitely generated. The factors of this series are finitely generated abelian groups. These are Noetherian by the structure theorem for finitely generated abelian groups.