What are some examples of infinite nonabelian groups?

Question

I can only think of the general linear groups GLn(R). What are some other examples, if any?

Answer 1

The bijections of an infinite set $X$.

Answer 2

Jorge mentioned the group of all permutations of an infinite set. This is an example of an automorphism group of a structure - namely, the structure which is just an infinite set, with no relations, operations, etc.

In general, automorphism groups are neither abelian nor finite. For example, the automorphism group of the field $\mathbb{C}$ has size continuum, and is very non-abelian. (By contrast, note that the field $\mathbb{R}$ has no nontrivial automorphisms!)

Similarly, structures built by "amalgamating" lots of small pieces in a sufficiently "generic" way (Fraisse limits and the Urysohn space come to mind) have lots of automorphisms, and their automorphism groups are generally non-abelian.

Answer 3

1. Take the group product $A\times B$ where $A$ is any non-Abelian group and $B$ is any infinite group.

2. On the set $\mathbb R^2$ let $(x,y)+^* (x',y')=(x+x', y+y'e^x).$ Then $(\mathbb R^2,+^*)$ is a non-Abelian group.

   It is also an orderable group: Let $(x,y)<^*(x',y')$ iff $[\;(x<x')$ or $(x=x'\land y<y')\;].$ Then $<^*$ is a linear order, and for all $p,q,r \in \mathbb R^2$ we have $p<^*q\implies ((p+^*r<^*q+^*r)\land (r+^*p<^*r+^*q)).$

3. For $2\leq n\in \mathbb N$ take the set of $n\times n$ matrices of real numbers (or of complex numbers) that are invertible in matrix multiplication.