Let $G$ a compact Hausdorff and totally disconnected topological group. Then every neighborhood of 1 contains an open normal subgroup of finite index in $G$. I need this lemma to prove that every compact Hausdorff and totally disconnected topological group is a profinite group. I am trying but I can not prove it. Any suggestions?
2026-03-25 07:46:26.1774424786
Existence of an open normal subgroup of a neighborhood of 1 in a compact Hausdorff and totally disconnected topological group
850 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GROUP-THEORY
- What is the intersection of the vertices of a face of a simplicial complex?
- Group with order $pq$ has subgroups of order $p$ and $q$
- How to construct a group whose "size" grows between polynomially and exponentially.
- Conjugacy class formula
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- A group of order 189 is not simple
- Minimal dimension needed for linearization of group action
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
- subgroups that contain a normal subgroup is also normal
- Could anyone give an **example** that a problem that can be solved by creating a new group?
Related Questions in TOPOLOGICAL-GROUPS
- Are compact groups acting on Polish spaces essentially Polish?
- Homotopy group of rank 2 of various manifolds
- A question on Group of homeomorphism of $[0,1]$.
- $G\cong G/H\times H$ measurably
- Is a connected component a group?
- How to realize the character group as a Lie/algebraic/topological group?
- Show $\widehat{\mathbb{Z}}$ is isomorphic to $S^1$
- a question on Ellis semigroup
- Pontryagin dual group inherits local compactness
- Property of the additive group of reals
Related Questions in PROFINITE-GROUPS
- $Ext(-,\mathbb Z)$ for profinite (Galois) groups
- Profinite completions and inverse limits
- Relation between profinite abelian groups and $\hat{\Bbb{Z}}$ modules
- Infinite Cyclotomic Extension of $\mathbb{Q}$
- Question about why inertia subgroup is a closed subgroup of the Galois group
- Conjugate subgroups in inverse limit of groups
- $H^1(G, \mathbb{Z}/p \mathbb{Z})$ and linearly independent elements in open subgroups.
- Example Powerful Pro-$p$-Groups
- Profinite completion of integers
- Every profinite group is naturally an affine group scheme over $\mathbb Q$?
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?
Lemma 3.9 in here.
If and when I have more time, I can copy the proof or you can DIY.