# Find an example of sets of cosets of different cardinality

Mathematics Asked on January 7, 2022

$$G$$ is a finite group. Let $$H$$ be a subgroup of $$G$$. Is there an example of $$G$$ and $$H$$ such that
$${rm Card}({Hxhmid hin H})neq{rm Card}({Hyhmid hin H}),$$
where $$x,yin Gsetminus H$$? Here $${rm Card}$$ means cardinality, namely the number of elements contained in a set, so I wonder if we can find two sets of cosets $${Hxhmid hin H}$$ and $${Hyhmid hin H}$$ such that the number of cosets contained in one set is different from the other one.

Could you give me some help? Thank you!

$$G:=S_5$$ and $$H:={(1),(23),(24),(34),(234),(243)}cong S_3$$. Set $$x:=(35)$$ and $$y:=(13)(45)$$. We have begin{align} &Hx={(35),(253),(24)(35),(345),(2534),(2453)},\ &Hy={(13)(45),(132)(45),(13)(254),(1354),(13542),(13254)}. end{align} The set of cosets $${Hxhmid hin H}$$ has $$3$$ elements and they are begin{align} &{(35),(253),(24)(35),(345),(2534),(2453)},\ &{(235),(25),(2435),(2345),(25)(34),(245)},\ &{(345),(2543),(2354),(45),(254),(23)(45)}; end{align} the set of cosets $${Hyhmid hin H}$$ has $$6$$ elements and they are begin{align} & {(13)(45),(132)(45),(13)(254),(1354),(13542),(13254)},\ &{(123)(45),(12)(45),(12543),(12354),(12)(354),(1254)},\ &{(13)(245),(13452),(13)(25),(13524),(1352),(134)(25)},\ &{(1453),(14532),(14253),(14)(35),(142)(35),(14)(253)},\ &{(14523),(1452),(143)(25),(14)(235),(14352),(14)(25)},\ &{(12453),(12)(345),(1253),(124)(35),(12)(35),(12534)}. end{align}

Hence we are done.

Answered by user792898 on January 7, 2022

Consider the symmetry group $$D_8$$ of a square. Let $$C_2$$ denote the subgroup fixing a corner $$x$$ (so $$C_2$$ consists of the identity, and reflection through the diagonal containing $$x$$). Then we can identify the $$4$$ corners of the square with the cosets of $$C_2$$. That is all the elements of $$C_2g$$ map $$x$$ to $$xg$$, so we may identify the coset $$C_2g$$ with the corner $$xg$$, for each $$gin D_8$$.

The orbits of corners under $$C_2$$ have different sizes: one orbit is the two corners adjacent to $$x$$, another is the single corner opposite to $$x$$.

Thus if $$a$$ is a $$90^circ$$ rotation, then $${C_2ah|hin C_2}$$ is two cosets, whilst $${C_2a^2h|hin C_2}$$ is one.

Answered by tkf on January 7, 2022

First consider $$G=S_3$$ and $$H={e,(1 2)}$$, with $$x=e$$ and $$y=(1 2 3)$$. Then $${Hxhmid hin H} = {H} quadtext{while}quad {Hyhmid hin H} = {H(1 2 3), H(1 2)}.$$

But wait, you say, we're not allowed to take $$xin H$$? This isn't actually that serious a restriction, since for any nontrivial group $$K$$ and any $$kin Ksetminus{e}$$, we can now replace $$G$$ by $$Gtimes K$$ and $$H$$ by $$Htimes{e}$$, and $$x$$ and $$y$$ by $$xtimes k$$ and $$ytimes k$$.

Answered by Greg Martin on January 7, 2022

## Related Questions

### Is it true that if $P(int_0^T f^2(s) ds<infty)=1$ then the exponential defines a density?

1  Asked on November 29, 2021 by user658409

### Random walk returning probability

3  Asked on November 29, 2021

### Prove that $mathrm{ht}(P/Ra)=mathrm{ht}(P) -1$

1  Asked on November 29, 2021

### Bayesian statistics notation: “$P(text{event}|x)$” vs “$P(text{event}|theta, x)$”

1  Asked on November 29, 2021

### Finding the third side of a triangle given the area

3  Asked on November 29, 2021

### Analytic continuation of $Phi(s)=sum_{n ge 1} e^{-n^s}$

1  Asked on November 29, 2021 by geocalc33

### If the monoid algebra $R[M]$ is finitely generated, then $M$ is a finitely generated monoid.

2  Asked on November 29, 2021 by dylan-c-beck

### Second cohomology group of an affine Lie algebra

0  Asked on November 29, 2021 by b-pasternak

### A problem on spectrum of a self-adjoint operator

1  Asked on November 29, 2021 by surajit

### Let $Lin End(V)$ with $L(V)=W$. Then $Tr(L)=Tr(L|_W)$

1  Asked on November 29, 2021

1  Asked on November 29, 2021

### Let $lambda$ be a real eigenvalue of matrix $AB$. Prove that $|lambda| > 1$.

1  Asked on November 29, 2021

### Integral of a product of Bessel functions of the first kind

1  Asked on November 29, 2021 by user740332

### Connectives in George Tourlakis’ Mathematical Logic

1  Asked on November 29, 2021 by darvid

### Let $T=int_{0}^{x}f(y)dt$. Find eigenvalues and range of $T+T^*$

0  Asked on November 29, 2021

### Find the smallest eigenvalue of $G=[ exp(-(x_i-x_j )^2]_{i,j}$ for ${bf x}=[x_1,dots,x_n]$

2  Asked on November 29, 2021

### Proof of convergence of $sum_{n=1}^{infty}frac{(-1)^{lfloor nsqrt{2}rfloor}}{n}$

1  Asked on November 29, 2021 by kubus

### Inverting product of non-square matrices?

1  Asked on November 29, 2021

### Half-SAT/ Half-Satisfiability

1  Asked on November 29, 2021 by f-u-a-s