TITLE: Group Theory via Category Theory
QUESTION [32 upvotes]: I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for someone who already knows category theory but doesn't know any group theory. I have been given the liberty to decide how I build the theory. I already have a vague idea of what is to be done.

However, I would like to hear ideas about what should be done. So I solicit advice on things I should emphasize, the ways I can exploit the given familiarity with category theory for a more economic presentation and/or a exposition through "the path of least resistance". And please feel free to also mention any tips or precautions.
Please refer me to material along this line.

Thanks!

REPLY [8 votes]: There are two books which link  category theory and group theory via the use of groupoids. A pioneer in this was Philip Higgins, whose 1971 van Nostrand Notes are now available as free download as Categories and Groupoids (T&G). Following Philip's lead I started investigating groupoids in $1$-dimensional homotopy theory in 1965, and concluded that the groupoid point of view made many things clearer. The book I published in 1968 is now in its third edition and available as  Topology and Groupoids.   A key tool was the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, chosen according to the geometry, which dates from 1967. The above are the only undergraduate  texts in English using this concept.  here is a  mathoverview discussion on many base points, and a  presentation on the background in algebraic topology was given at Galway in  December, 2014. See also my preprint page.
A feature of both of these books is the use of the following construction. Let $G$ be a groupoid with object set $X$ and let $f:X \to Y$ be a function to a set $Y$. Then we obtain a new groupoid which can be written $U_f(G)$ with object set $Y$ and a nice universal property. From this can be deduced the construction of coproducts of groups, and of free groups and free groupoids. This goes a long way to covering point 2 of @goblin's comment. Thus there is a combinatorial groupoid theory which in principle includes combinatorial group theory.
January 23, 2017 I should say that "combinatorial groupoid theory" is my own coinage, but also reflects JHC Whitehead's term "combinatorial homotopy theory". 
The areas include those mentioned above and also fibrations and covering morphisms, groups acting on groupoids, (see T&G for all these). Other references are in my groupoids web page. 

To answer the comment of goblin: how does one obtain a groupoid from a bundle of groups? The latter can be seen as a disjoint family $\{G_s: s \in S\}$ of groups. 
The associated groupoid $G$ has object set $S$ and a group $G_s$ for each $S \in S$, so that $G(s,t)$ is empty if $s \ne t$.
Another feature of the category of groupoids  is the variety of morphisms. For groups there are mainly two  special kinds of morphisms: injective or surjective. For groupoids we also have fibrations, covering morphisms, universal morphisms, orbit morphisms.  
There is a special groupoid $\mathcal I$ which has two objects $0,1$ and only one non identity $\iota: 0 \to 1$, and so also only $\iota^{-1}: 1 \to 0$. This seems a completely useless kind of object! But it defines a homotopy theory for groupoids, and also  plays the same role in the category of groupoids as does the integers $ \mathbf Z$ in the category of groups. Further, identifying $0,1$ in $\mathcal I$ in the category of groupoids gives the integers $\mathbf Z$, which explains the fundamental group of the circle!