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} $.