Classification of groups: finite, countable, continuous

Question

I'm trying to follow a set of lecture notes on group theory. They make a note that groups can fall into three categories: finite, countable, and continuous. The first two I am fine with: they rely on the cardinality of the underlying set. The third typing is "continuous groups." The examples given are $\mathbb{R}$ (with addition), $\mathbb{C}$ (with addition), $\mathrm{GL}_n (\mathbb{R})$ (with matrix multiplication), and $S^1$ (the complex unit circle with multiplication).

I am trying to understand what they mean by "continuous groups." One definition I read was that the operations must be continuous, but then I don't understand why $\mathbb{Z}$ and $\mathbb{Q}$ are not included, as addition should be continuous no matter the set. If it means that the underlying set is uncountable, then I don't understand why, say, $\mathrm{SL}_n (\mathbb{R})$ or $\mathrm{SL}_n (\mathbb{C})$ are not included. I read that continuous groups have to be infinite, but I do not understand why.

I'd appreciate if someone could clarify this.

Answer 1

The most direct interpretation of "continuous groups" is topological groups: that is, groups which are also topological spaces in which the multiplication operation is continuous. This is unsatisfying because any group can be a topological group with the discrete or indiscrete topology. Even other topologies don't lend to an intuition of "continuous" the way the matrix group examples do, for example $(\mathbb{Z},+)$ is a topological group with the arithmetic progression topology. (All topologies on a finite group $G$ are those which are discrete modulo a normal subgroup; see here.)

This is observable even in uncountable topological groups, namely profinite groups (which include the profinite integers, $p$-adic integers for any prime $p$, the absolute Galois group of the rationals, i.e. the symmetries of all algebraic numbers, and automorphism groups of infinite trees). Since these groups are totally disconnected, they are at the extreme from Lie groups, which are groups with have a compatible manifold structure (which, turns out, is WLOG analytic). Crazily, you can even merge these two opposing worlds together, as you might see with the adeles or solenoids.

These things are important in number theory and are perfectly valid topological groups, but arguably do not convey the intuition of "continuous" the author originally had in mind. Either way, though, there are plenty of uncountable groups which the author would not call "continuous," and it is easy to drum up more artificial examples (like $\mathbb{Z}_2^{\oplus\mathfrak{c}}$ or $\mathrm{Sym}\,\mathbb{N}$), so their classification is incomplete.

Lie groups is probably the best interpretation of what the author meant, given their examples. And it would include $\mathrm{SL}_n$; I don't know why you think the author wouldn't include that, I think they would. (Notably, there do exist Lie groups which are not "linear," i.e. they have no faithful representation by matrices; the go-to examples are the double cover $\mathrm{Mp}_2$ (the metaplectic group) and universal cover $\widehat{\mathrm{SL}_2\mathbb{R}}$ of $\mathrm{SL}_2\mathbb{R}$.)

Answer 2

This would need some more context to be able to answer, or a full quote from the lecture notes.

In any case I disagree that all groups are finite, countable, or "continuous". There are certainly infinite groups which are not "continuous" in any reasonable sense.

Answer 3

When we have a group with the Set of Elements $S$ & the Binary Operation $+$ , we have to look at the elements in $S$ to classify the group.

When $S$ is finite , having $N$ elements , then it is a group of order $N$ , eg $S=\{0,1,2\}$ , Integers MOD 3.

When $S$ is infinite , having countable elements , then it is a countable group , eg Integers.

When $S$ is infinite , having uncountable elements & the elements are "continuous" , then it is a continuous group , eg real numbers or complex numbers.

Perspective of the Set $S$ & the Continuous Mapping is used to classify the group.

There are groups which are outside these 3 classes.

Some references :

https://mathworld.wolfram.com/ContinuousGroup.html
"A group having continuous group operations. A continuous group is necessarily infinite, since an infinite group just has to contain an infinite number of elements. But some infinite groups, such as the integers or rationals, are not continuous groups."

https://webhome.weizmann.ac.il/home/fnkirson/Alg15/8.Continuous_groups.pdf
".... In general, the elements of a continuous group may be labelled by a number of continuous variables. ...."

http://hep.spbu.ru/images/lectures/method/ioffe1.pdf
"INTRODUCTION TO CONTINUOUS GROUPS"

http://sces.phys.utk.edu/~moreo/mm08/Jessica.pdf
"Credit for introducing continuous groups into literally all branches of math is mainly due to the work of mathematicians Sophus Lie and Felix Klein.
Lie is considered to be one of the last great mathematicians of the 19th century, and continuous groups are now more commonly known as Lie groups."

http://ckw.phys.ncku.edu.tw/public/pub/Notes/Mathematics/GroupTheory/Tung/pdf/6._1DContinuousGroups.pdf
"Continuous groups consist of group elements labelled by one or more continuous variables, .... where each variable has a well-defined range."

https://en.wikipedia.org/w/index.php?title=Continuous_group&redirect=no
getting redirected to https://en.wikipedia.org/wiki/Topological_group

In short inaccurate summary , we can say that the Mapping $+$ is Continuous when , with $X$ in the neighbourhood of $x$ & $Y$ in the neighbourhood of $y$ , we have $X+Y^{-1}$ in the neighbourhood of $x+y^{-1}$