## Theory of gr.group – Measured in cosets in a group?

This should be possible for any group, assuming I have data on the correct extent of the measures.

Leave $$mathcal {B}$$ be the Boolean algebra generated by cosets of finite index subgroups $$G$$. Then there is a finite additive (in fact $$G$$-invariant) measure $$mu$$ in $$G$$ so that if $$C$$ It is a coset of $$H$$Y $$[G:H]= n < infty$$, so $$mu (C) = 1 / n$$. Specifically, if $$A in mathcal {B}$$ then there is a finite index subgroup $$H$$ of $$G$$, such that $$A$$ is a union of cosets of $$H$$. Yes $$n$$ is the index, and $$m$$ is the number of cosets in the union then it is established $$mu (A) = m / n$$. You can directly verify that this is a well-defined measure as desired.

Now one must be able to extend $$mu$$ arbitrarily to a finely additive measure in $$mathcal {P} (G)$$ (which will not be $$G$$-invariant necessarily). I think this follows from Section 457 of Fremlin's "Theory of Measure".

Observation 1: the initial measure $$mu$$ it is in fact the only $$G$$-invariant finite probability measure additive in $$mathcal {B}$$, and can be constructed from Haar's measure in the profinite completion of $$G$$. In particular, if $$mathcal {N}$$ Denotes the collection of normal subgroups of finite index of $$G$$, then the profinite ending is $$hat {G} = varprojlim _ { mathcal {N}} G / N$$. We can write elements of $$hat {G}$$ as $$(C_N) _ {N in mathcal {N}}$$, where $$C_N$$ It is a coset of $$N$$. Given a set $$A$$ in $$mathcal {B}$$define $$X_A$$ to be the set of $$(C_N) _ {N in mathcal {N}} in hat {G}$$ such that $$C_N cap A neq emptyset$$ for all $$N en mathcal {N}$$. So $$X_A$$ is closed and you can check that $$mu (A)$$ Make the measure of $$X_A$$.

Observation 2: Perhaps it is also worth mentioning that if $$G$$ is susceptible (for example, abelian) then there is a $$G$$-invariant finite probability measure additive in $$mathcal {P} (G)$$, which must satisfy the desired conditions directly by finite additivity.

## nt.number theory – Double Cosets and Weber's function

Leave $$n$$ be an odd positive integer leave $$mathcal M_n$$ is the set of all $$2$$-by-$$2$$ Primitive matrices with integral entries and with determinant. $$n$$.

Leave $$Gamma$$ be the subgroup of $$operatorname {SL} _2 ( mathbb Z)$$ generated by the matrices $$T ^ 2 = begin {pmatrix} 1 & 2 \ 0 & 1 end {pmatrix}$$ Y $$S = begin {pmatrix} 0 & -1 \ 1 & 0 end {pmatrix}$$.

So
$$Gamma = bigg lbrace begin {pmatrix} a & b \ c & d end {pmatrix}: begin {pmatrix} a & b \ c & d end {pmatrix} equiv begin {pmatrix} 1 & 0 \ 0 & 1 end {pmatrix} text {or} begin {pmatrix} a & b \ c & d end {pmatrix} equiv begin pmatrix} 0 & 1 1 & 0 end {pmatrix} text {mod} 2 bigg rbrace.$$

How many cosets are there in $$Gamma backslash mathcal M_n / Gamma$$ ?

Leave $$r, s, t$$ be positive integers suppose that $$rt = n$$, $$s <2t$$, cast $$s$$ even. There are matrices $$A, B in Gamma$$ such that $$A begin {pmatrix} n & 0 \ 0 & 1 end {pmatrix} B = begin {pmatrix} r & s \ 0 & t end {pmatrix}$$?

Motivation.

The Hauptmodul for the group. $$Gamma$$ is the function
$$mathfrak f ( tau) ^ {24} = q ^ {- 1/2} prod_ {k = 1} ^ { infty} (1 + q ^ {n-1/2}).$$
Leave $$Phi_n (X)$$ be the minimum polynomial of $$mathfrak f (n tau) 24$$ finished $$mathbb C ( mathfrak f ^ {24})$$. Is $$mathfrak f left ( frac {r tau + s} {t} right)$$ a root of $$Phi_n (X)$$?

## Can the double cosets of the cyclic subgroups be separated into a special linear group?

Leave $$A, B in mathrm {SL} _3 ( mathbb {Z})$$. Set
$$S = langle A rangle cdot langle B rangle = {A ^ mB ^ n: m, n in mathbb {Z} }.$$

Is $$S$$ closed in the proficient topology in
$$mathrm {SL} _3 ( mathbb {Z})$$ ?

Equivalently (using the property of the congruence subgroup), I am asking if for each $$C in mathrm {SL} _3 ( mathbb {Z})$$ for which $$C equiv A ^ {m_k} B ^ {n_k} pmod k$$ have for any $$k$$, we necessarily have $$C = A ^ mB ^ n$$ for some $$m, n in mathbb {Z}$$.