Linear group

Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem.

They arise as linear algebraic groups, that is, as subgroups of GLn defined by a finite number of equations.

A finite group G of order n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of G on the group ring K[G] by left (or right) multiplication is linear and faithful.

While example 4 above is too general to define a distinctive class (it includes all linear groups), restricting to a finite index set I, that is, to finitely generated groups allows to construct many interesting examples.

For example: In some cases the fundamental group of a manifold can be shown to be linear by using representations coming from a geometric structure.

Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above.