I've read the textbook Groups and Their Graphs by Grossman, and I'm interested in learning more about graphs. I know about O. Ore's book in the same series (Graphs and Their Uses), but I'm interested in a book which will tell me more about the relation between graphs and groups. I don't know any advanced mathematics (I only about group theory and graph theory from Grossman's book), so please recommend books which are not too complicated. I would also be glad for recommendations of books which are about group theory but have a focus on graphs.
[Math] Textbooks on graph theory
book-recommendationgraph theorygroup-theoryreference-request
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The book categories for the working mathematician, by Saunder's MacLane, comes well-recommended. Although I have never quite got around to reading it, it has such a promising title! Note that Saunder's MacLane was one of the original category theorists.
For an explicit example of graph theory and category theory working together to prove results on groups, there is a well-known link between the category of graphs and subgroups of free groups. The classical reference is Topology of Finite Graphs by Stallings. For example, if the non-diagonal component of the fibre product of a graph with itself is simply connected then the corresponding subgroup is malnormal. Dani Wise lifted this idea to the more general category of "cubical complexes", which allowed him to prove some famous open problems in group theory and G&T (for example, he proved the virtually Haken conjecture, and that every one-relator group with torsion is residually finite). I found Wise's paper The residual finiteness of positive one-relator groups to be especially helpful.
You say you want to use category theory and graphs to look at endomorphisms of groups. Well, one of the questions I was attacking in my PhD thesis was the following.
Fix a class of groups $\mathcal{C}$. Does every (countable) group occur as the outer automorphism group of a group from the class $\mathcal{C}$?
I fixed a class $\mathcal{C}$, and I managed to prove that the above result held for this class so long as I could prove that a certain group (in reality, a class of very similar groups) had a malnormal subgroup (with certain additional properties). So, I then took the fibre product of a "subgroup" in the ambient free group and (with a bit of effort) proved that its malnormality fell down to my group.
This doesn't quite answer your question, but I thought it relevant enough to mention. If you want, I can send you a copy of my thesis.
$\newcommand \G{\mathbf{G}}$ $\newcommand \GL{\mathrm{GL}}$ $\newcommand \SL{\mathrm{SL}}$ $\newcommand \Z{\mathbb{Z}}$ $\newcommand \Q{\mathbb{Q}}$ Just to get this out of the unanswered queue, let me give the following answer:
1) Yes! There are actually tons. Any book on 'linear algebraic groups' will cover what you want (although Waterhouse's book is strange it sidesteps a lot of the theory). Specifically though, I would recommend these notes of Milne--I think they are about as good as one could possibly hope for in terms of completeness. They have the downside of being somewhat cagey about using modern algebraic geometry, but only in language. Namely, Milne does talk about nilpotents (which are pivotal in the characteristic $p$ theory since such simple maps like $\text{SL}_p\to\text{PGL}_p$ have non-reduced kernel) but he insists of talking about $\text{MaxSpec}(A)$ instead of $\text{Spec}(A)$ for some unbeknownst reason--it doesn't make a difference, but it's worth noting.
I am also always eager to rep Brian Conrad's notes (for most things). Specifically, there are these from a first course on linear algebraic groups. They have the strong plusses of being written from a 'modern algebro-geometric viewpoint' (e.g. quotients are defined as quotient fppf sheaves, which is what they should be defined as), but falls prey to the neuroticism that befalls all live-TeXed notes--they're fairly all over the place. Unfortunately, that doesn't really cover what you're asking for because it doesn't cover the structure theory of reductive groups (and their representations). For that you'll have to look at notes from his follow-up course. Again, these are probably my favorite notes for the topic, but are somewhat hard to quickly look things up in (something Milne's notes excel in).
Finally, if you want a 'quick fix' you can look at the first (~50 page) section of these notes of Brian Conrad--I think the same material is roughly contained in the appendix to his book (with Prasad and Gabber) Pseudo-reductive Groups.
Let me just note that if you're really only interested in the groups you've mentioned (over $\mathbb{Q}$), then everything is easily describable.
For $\mathbf{G}_m$ (everything in here will be over $\mathbb{Q}$) representations $\mathbf{G}_m\to\GL(V)$ correspond to $\Z$-gradings $\displaystyle V=\bigoplus_{i\in\Z}V_i$ where $\G_m$ acts on $V_i$ by the character $z\mapsto z^i$. In particular, every representation is semisimple and the simple representations are just characters (which are precisely the maps $z\mapsto z^i$).
For $\SL_2$ things are ever so slightly more complicated. Namely, $\SL_2$ is reductive (in fact, semisimple) and thus every algebraic representation of $\SL_2$ is semisimple. Thus, we really only need to describe the simple representations of $\SL_2$. To do this, note that $\SL_2$ naturally acts on homogenous polynomials of degree $m$ in the variables $x,y$ (by $(a_{ij})f(x,y):=f(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$, call this representation $V_m$. Then, every simple representation of $\SL_2$ is isomorphic to precisely one of these $V_m$.
(Note that it's not a coincidence that $V_m\cong \mathcal{O}(m)(\mathbb{P}^1)$ since $\mathbb{P}^1$ is the flag variety $\SL_2/B_2$ associated to $\SL_2$ and the Borel-Weil theorem describes a precise relation between representations of $\SL_2$ and (certain) line bundles on the flag variety--this works more genenerally for a semisimple group).
Finally, $\GL_2$ is also reductive (but not semisimple), thus to describe its representation theory we need only describe its simple representations. These come in two flavors. Namely, there is the irreducible tautological representation of $\GL_n$ (acting on $k^n$ in the usual way) and there are the representations $\wedge^i(k^n)$ for $i\leqslant n$. That's all of them.
Of course, to see where these came from--the real answer is the theory of dominant weights. That said, it's always easier to think about the analogy you know from the representation theory of a finite group. Namely, if $G$ is a finite group and $\text{Reg}(G)$ denotes the left regular representation of $G$, in other words $G$ acting on the group algebra $\mathbb{C}[G]$, then there is a decomposition
$$\text{Reg}(G)\cong\bigoplus_{\rho\in\text{Irr}(G)}\rho^{\dim\rho}$$
where $\text{Irr}(G)$ is the set of isomorphism classes of irreducible algebraic representations of $G$. A similar thing happens for a reductive group $G$. For example, for $\GL_n$ one has that the coordinate ring of $\GL_n$ (as a variety) decomposes as a $\GL_n\times\GL_n$-rep (via the multiplication map) as a direct sum of $V\boxtimes V^\ast$ where $V$ runs over the irreducible representations of $\GL_n$. So, see if you can use this to sort out what happens at least for $\G_m$.
2) As I mentioned in the comment, this really depends on what $\Q$-algebras you pick. For example, you know from the Yoneda philosophy that any representation $\rho:G\to\GL_n$ of an algebraic group $G$ is determined by the group maps $G(R)\to\GL_n(R)$ as $R$ varies over all $\Q$-algebras. In, fact it suffices to think about it for $R=\mathcal{O}_G(G)$ which gives you the notion of a comodule.
Of course, things are going to be bad if you want to consider just the values of the representation on some random $R$, even for examples $R=\Q$ I think. I don't have an example off-hand to be honest (there's probably not a hard one) but if you look anything other than $\Q$, say $\Q(i)$, you'll get lots of representations of $\GL_2(\Q(i))$ (say) that won't be algebraic--think about the one that is induced from the non-trivial automorphism of $\Q(i)$.
Best Answer
You might be interested in the book Graphs, Groups and Trees by John Meier. It is a very readable introduction to "geometric group theory", and is pitched at "advanced undergraduate" level (in the question linked in the comments, some of the books are graduate level and above). Geometric group theory can be interpreted as "the study of groups using their actions", and the simplest actions are actions on graphs. Hence, this book studies groups by using their actions on graphs.
Topics covered in the book include group actions, Cayley graphs (every group acts on a graph, and the Cayley graph is such a graph), actions on trees and basic Bass-Serre theory, the word problem for groups, regular lagnauges and normal form, and the coarse geometry of groups. There are lots of examples to get your teeth into (every other chapter takes a specific group and analyses it using the previous chapter).
I should say that Meier's book assumes a working knowledge of group theory, but I would be surprised if there existed a book on this subject which did not!
@ABajaj The book you were reading, by Grossman and Magnus, was from the "new mathematical library". This "library" was a collection of books pitched at your level (US high school) which accompanied a new method of teaching maths (called new math) in the US. The method was generally considered a failure, and therefore I would doubt if there was a set follow-on book. Meier's book perhaps requires a small jump from where you are just now, but this jump can most likely be helped by also buying an introductory group theory text to use as a reference. If you know what a normal subgroup is, then you are probably good to go! (The word "Sylow" does not enter Meier's book, although enters every single "standard" group theory text, so your jump will not be too big!)
I should point out that the name of Magnus is a famous one in geometric group theory. So Meier's book, and geometric group theory in general, is a natural place to go next.