If $H$ is subgroup of $G$ such that the index of $H$ in $G$ is $n$ and $\pi_H$ is the permutation representation of the action of $G$ on the left cosets of $H$, is $\pi_H$ a map from $H$ to $S_n$? I am a bit confused about how $\pi_H$ relates to the symmetric group.
2026-04-03 14:30:53.1775226653
Defining a map based on a group action on left cosets
251 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 SYMMETRIC-GROUPS
- Orbit counting lemma hexagon
- A "Restricted Sudoku" Symmetry Group Question
- Show, by means of an example, that the group of symmetries of a subset X of a Euclidean space is, in general, smaller than Sym(x).
- Prove that $\sigma$ is a power of $\tau$ when they commute $\sigma\tau=\tau\sigma$.
- Proof verification - the only group of order 24 without normal sylow subgroup is $S_4$.
- Symmetry subgroup of a cube
- Subgroup generated by $S$ is $A_5$
- Question about semigroups of permutations
- Symmetry of the tetrahedron as a subgroup of the cube
- Interpretation of wreath products in general and on symmetric groups
Related Questions in GROUP-ACTIONS
- Orbit counting lemma hexagon
- Showing a group G acts on itself by right multiplication
- $N\trianglelefteq G$, $A$ a conjugacy class in $G$ such that $A\subseteq N$, prove $A$ is a union of conjugacy classes
- Show that the additive group $\mathbb{Z}$ acts on itself by $xy = x+y$ and find all $x\in\mathbb{Z}$ such that $xy = y$ for all $y\in\mathbb{Z}$.
- Number of different k-coloring of an $n\times m$ grid up to rows and columns permutations
- How to embed $F_q^\times $ in $S_n$?
- orbit representatives for the group of unipotent matrix acting on the set of skew-symmetric matrices
- $S_n$ right-action on $V^{\otimes n}$
- Interpretation of wreath products in general and on symmetric groups
- Regarding action of a group factoring through
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?
The map $\pi_H$ is technically a homomorphism from $G$ to $S_X$ where $X = G/H$. Now if $G$ has finite index, then $|X| = n$, and so $S_X \cong S_n$. Under this isomorphism you can think of $\pi_H$ as a homomorphism from $G$ to $S_n$.