On Defining the Fourier Transform and Performing Changes of Variable on Quotient Subgroups of $\mathbb{Q}/$

Much to my dismay, in my work the more number-theoretic side of harmonic analysis (ex: the fourier transform on the adeles, on the profinite integers, etc.), I have found myself struggling with technicalities that emerge from frustratingly simple issues—so simple (and yet, so technical) that I haven’t been able to find anything that might shine any light on the matter.

Let $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ be the space of functions $ f:\mathbb{Q}\rightarrow\mathbb{C}$ which satisfy $ \sum_{t\in T}\left|f\left(t\right)\right|^{2}<\infty$ for all bounded subsets $ T\subseteq\mathbb{Q}$ . Letting $ \mu$ be any positive integer, it is easy to show that any functions which is both $ \mu$ -periodic (an $ f:\mathbb{Q}\rightarrow\mathbb{C}$ such that $ f\left(t+\mu\right)=f\left(t\right)$ for all $ t\in\mathbb{Q})$ and in $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ is necessarily an element of $ L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ , the complex hilbert space of functions $ f:\mathbb{Q}/\mu\mathbb{Z}\rightarrow\mathbb{C}$ so that: $ $ \sum_{t\in\mathbb{Q}/\mu\mathbb{Z}}\left|f\left(t\right)\right|^{2}<\infty$ $ Equipping $ \mathbb{Q}/\mu\mathbb{Z}$ with the discrete topology, we can utilize Pontryagin duality to obtain a Fourier transform: $ \mathscr{F}_{\mathbb{Q}/\mu\mathbb{Z}}$ . The ideal case is when $ \mu=1$ . There, $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}$ is an isometric hilbert space isomorphism from $ L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)$ to $ L^{2}\left(\overline{\mathbb{Z}}\right)$ , where: $ $ \overline{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$ $ is the ring of profinite integers, where $ \mathbb{P}$ is the set of prime numbers, and where $ L^{2}\left(\overline{\mathbb{Z}}\right)$ is the space of functions $ \check{f}:\overline{\mathbb{Z}}\rightarrow\mathbb{C}$ which are square-integrable with respect to the haar probability measure $ d\mathfrak{z}=\prod_{p\in\mathbb{P}}d\mathfrak{z}_{p}$ on $ \overline{\mathbb{Z}}$ .

The first sign of trouble was when I learned that, for any integers $ \mu,\nu$ , the (additive) quotient groups $ \mathbb{Q}/\mu\mathbb{Z}$ and $ \mathbb{Q}/\nu\mathbb{Z}$ are group-isomorphic to one another, and thus, that both have $ \overline{\mathbb{Z}}$ as their Pontryagin dual. From my point of view, however, this isomorphism seems to only cause trouble. In my work, I am identifying $ L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ with the set of $ \mu$ -periodic functions $ f:\mathbb{Q}\rightarrow\mathbb{C}$ which are square integrable with respect to the counting measure on $ \mathbb{Q}\cap\left[0,\mu\right)$ . As such, $ \mathbb{Q}/\mu\mathbb{Z}$ and $ \mathbb{Q}/\nu\mathbb{Z} $ cannot be “the same” from my point of view, because $ f\left(t\right)\in L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ need not imply that $ f\left(t\right)\in L^{2}\left(\mathbb{Q}/\nu\mathbb{Z}\right)$ .

In my current work, I am dealing with a functional equation of the form:$ $ \sum_{n=0}^{N-1}g_{n}\left(t\right)f\left(\frac{a_{n}t+b_{n}}{d_{n}}\right)=0$ $ where $ N$ is an integer $ \geq2$ , where the $ g_{n}$ s are known periodic functions, where $ f$ is an unknown function in $ L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)$ (i.e., $ f\left(t\right)\in L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ and $ f\left(t+1\right)=f\left(t\right)$ for all $ t\in\mathbb{Q})$ , and where $ a_{n},b_{n},d_{n}$ are integers with $ \gcd\left(a_{n},d_{n}\right)=1$ for all $ n$ . For brevity, I’ll write: $ $ \varphi_{n}\left(t\right)\overset{\textrm{def}}{=}\frac{a_{n}t+b_{n}}{d_{n}}$ $ Because $ \varphi_{n}\left(t+1\right)$ need not equal $ \varphi_{n}\left(t\right)+1$ , the individual functions $ f\circ\varphi_{n}$ , though periodic and in $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ , are not necessarily going to be of period $ 1$ . Letting $ p$ denote the least common multiple of the periods of $ g_{n}$ and the $ f\circ\varphi_{n}$ s, I can view the functional equation as existing in $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ , and as such, I hope to be able to simplify it by applying the fourier transform.

Letting $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ denote the duality pairing between elements $ t\in\mathbb{Q}$ (or $ \mathbb{Q}/\mu\mathbb{Z}$ ) and $ \mathfrak{z}\in\overline{\mathbb{Z}}$ , the idea is to multiply the functional equation by $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ , sum over an appropriate domain of $ t$ (ideally, $ \mathbb{Q}/p\mathbb{Z}$ ), make a change of variables in $ t$ to move one of the $ \varphi_{n}\left(t\right)$ s out of $ f$ and into $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ , pull out terms from this character, and then invert the fourier transform to return to $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ with a vastly simpler equation. My main difficulty can be broken into three parts:

(1) Is taking the least common multiple of the periods to reformulate the functional equation as one over $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ legal?

(2) I know that the fourier transform $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}:L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)\rightarrow L^{2}\left(\overline{\mathbb{Z}}\right)$ is given by:$ $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{t\in\mathbb{Q}/\mathbb{Z}}f\left(t\right)e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ $ and that the inverse transform is: $ $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}^{-1}\left\{ \check{f}\right\} \left(t\right)\overset{\textrm{def}}{=}\int_{\overline{\mathbb{Z}}}\check{f}\left(\mathfrak{z}\right)e^{-2\pi i\left\langle t,\mathfrak{z}\right\rangle }d\mathfrak{z}$ $ However, I am at a loss as to what formula to use for $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}$ and its inverse, and for two reasons. On the one hand, because $ \mathbb{Q}/\mathbb{Z}$ and $ \mathbb{Q}/p\mathbb{Z}$ are group-isomorphic, what is to stop me from using the same formula for their fourier transforms? On the other hand, if I use a modified formula—say:$ $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{t\in\mathbb{Q}/p\mathbb{Z}}f\left(t\right)e^{2\pi i\left\langle \frac{t}{p},\mathfrak{z}\right\rangle }$ $ does the fact that $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)\in L^{2}\left(\overline{\mathbb{Z}}\right)$ then mean that I can recover $ f$ by applying $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}^{-1}$ , or do I have to also modify it in order to make everything consistent? Knowing the correct formula for the fourier transform on $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ and its inverse is essential.

(3) I would like to think that performing a change-of-variables for a sum of the form:$ $ \sum_{\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)$ $ (where $ \alpha,\beta,r\in\mathbb{Q}$ , with $ \alpha\neq0$ and $ r=\frac{p}{q}>0$ ) would be a relatively simple matter, but, that doesn’t appear to be the case. For example, if $ t\in\mathbb{Q}\rightarrow f\left(\alpha t+\beta\right)\in\mathbb{C}$ is not a $ r$ -periodic function, then this sum is not well-defined over the quotient group $ \mathbb{Q}/r\mathbb{Z}$ . Worse yet—supposing $ f$ is $ r$ -periodic take a look at this: write elements of $ \mathbb{Q}/r\mathbb{Z}$ in co-set form: $ t+r\mathbb{Z}$ , where $ t\in\mathbb{Q}$ . Then, make the change-of-variable $ \tau=\alpha t+\beta$ . Consequently, the set of all $ \tau$ is:$ $ \alpha\left(\mathbb{Q}/r\mathbb{Z}\right)+\beta=\left\{ \alpha\left(t+r\mathbb{Z}\right)+\beta:t\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:t\in\mathbb{Q}\right\}$ $ . Here is where things get loopy.

(1) Since $ \alpha,\beta\in\mathbb{Q}$ with $ \alpha\neq0$ , the map $ \varphi\left(t\right)\overset{\textrm{def}}{=}\alpha t+\beta$ is a bijection of $ \mathbb{Q}$ . As such, I would think that:$ $ \left\{ \tau+\alpha r\mathbb{Z}:t\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:\varphi^{-1}\left(\tau\right)\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:\tau\in\varphi\left(\mathbb{Q}\right)\right\}$ $ and hence:$ $ \alpha\left(\mathbb{Q}/r\mathbb{Z}\right)+\beta=\left\{ \tau+\alpha r\mathbb{Z}:\tau\in\mathbb{Q}\right\} =\mathbb{Q}/\alpha r\mathbb{Z}$ $ Using this approach, I obtain:$ $ \sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=\sum_{\tau\in\mathbb{Q}/\alpha r\mathbb{Z}}f\left(\tau\right)$ $

(2) Since $ r=\frac{p}{q}$ , decompose $ \mathbb{Z}$ into its equivalence classes mod $ q$ :$ $ \left\{ \tau+\alpha r\mathbb{Z}:\tau\in\mathbb{Q}\right\} =\bigcup_{k=0}^{q-1}\left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\}$ $ and so:$ $ \sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=\sum_{k=0}^{q-1}\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau+\alpha rk\right)$ $ On the other hand, for each $ k$ :$ $ \left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\} =\left\{ \tau+\alpha rk+\alpha rq\mathbb{Z}:\tau\in\mathbb{Q}\right\}$ $ and, since $ \tau\mapsto\tau+\alpha rk$ is a bijection of $ \mathbb{Q}$ , the logic of (1) would suggest that:$ $ \left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\} =\left\{ \tau+\alpha rq\mathbb{Z}:\tau\in\mathbb{Q}\right\}$ $ for all $ k$ . But then, that gives:$ $ \sum_{k=0}^{q-1}\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau+\alpha rk\right)=\sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=q\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau\right)$ $ which hardly seems right.

Frechet Lie groups and their subgroups

1) Let $ G$ be a Fréchet Lie group. Let $ H$ be a closed subgroup. Is it always true that the centraliser of $ H$ is a Fréchet subgroup of the lie group?

2) Is the closed subgroup theorem valid for Fréchet Lie groups?

I’m unaware of the the literature in this field and would appreciate if someone could point me to some compendium of results regarding Fréchet Lie groups.

Frechet Lie groups and their subgroups

1)Let $ G$ be a Frechet lie group. Let $ H$ be a closed subgroup. Is it always true that the centraliser of $ H$ is a Frechet subgroup of the lie group?

2)Is the closed subgroup theorem valid for Frechet Lie groups?

I’m unaware of the the literature in this field and would appreciate if someone could point me to some compendium of results regarding Frechet Lie groups.

Permutations covered by subgroups?

Given integer $ m\in[1,n]$ fix a set $ \mathcal T$ of permutations in $ S_n$ . There are subgroups $ G_1,\dots,G_m$ of $ S_n$ so that $ \mathcal T$ is covered by cosets of $ G_1,\dots,G_m$ .

  1. My problem then is given $ \mathcal T$ is there always an $ m=O(poly(n))$ such that there are elements $ g_1,\dots,g_m\in S_n$ and some subgroups of $ G_1,\dots,G_m$ of $ S_n$ such that

$ $ \mathcal T\subseteq\cup_{i=1}^mg_iG_i$ $ $ $ (\sum_{i=1}^m|G_i|-|\mathcal T|)^2<m’$ $ where both $ m$ and $ m’$ are $ O(poly(n))$ .

  1. If not what is the trade off between $ m$ and $ m’$ ?

  2. Is it possible to get at least $ O(subexp(n))$ for both?

  3. If $ m’=0$ is there always a minimum $ m$ for all $ \mathcal T$ ?

Open normal subgroups of a profinite group form a basis for the open neighborhoods of 1

In the following link: An equivalent definition of the profinite group
I’m having trouble understanding the following quote from the answer:

So if you replace each subgroup in a basis by the intersection of its conjugates, you obtain a basis made up of normal subgroups.

How can one prove that these normal subgroups form a basis?

Is there another name for Goursat’s Lemma on subgroups of a direct product of groups?

I’m having trouble finding a textbook that discusses Goursat’s Lemma on subgroups of a direct product of groups. I’ve looked in several standard Algebra textbooks and I’ve only seen it in Serge Lang’s “Algebra” as an exercise.

Is it more commonly known by another name, or perhaps subsumed by a more commonly-taught theorem?

Bonus points, but not required: if not, why isn’t it included in these texts? The direct product is one of the standard first constructions and it seems like one of the first questions one would ask is “what is known about the subgroups of $ G \times H$ ?”

Find the amount of subgroups of order $3$ and $21$ in non-cyclic abelian group of order $63$

Find the amount of subgroups of order $ 3$ and $ 21$ in non-cyclic abelian group of order $ 63$ .

In first case I found the amount of elements that have order $ 3$ – there are $ 8$ of them, in second case there are $ 48$ elements of order $ 21$ . How do I connect these values with the amount of subgroups now?

Number of subgroups of an abelian p-group

Let $ p$ be a prime number and let $ n\in \mathbb{N}$ . I know that every abelian group of order $ p^n$ is uniquely a direct sum of cyclic groups of order $ p^{\alpha_i}$ where $ \sum \alpha_i = n$ . Now the question:

Among all abelian groups of order $ p^n$ which one has the most number of subgroups? Actually, I am looking for the Max number of subgroups so a close upper bound for the maximum number of subgroups would also be appreciated.