1) . These are cyclic groups. They are the group of i.e. the "integers modulo ". Of course, .
2) . These are permutation groups. A permutation on is a function which is a bijection. It is not hard to show . This group is not abelian if . More deep properties include that is not solvable. Called "permutation group".
3) . Any element in can be written as a product of -cycles. It can be shown that if a permutation in is a product of even # of -cycles then any other such -cycle decomposition must have even # of cycles. Therefore, it is well-defined to say a permuation is even if it can be decomposed into an even # of -cycles. This group is a subgroup of . Furthermore, it is a normal subgroup of . It is also simple for . Finally . Called "alternating group".
4) . Let be a finite field and . The set of all invertible matrices with coefficients in under multiplication will form this group. This is the "general linear group". Sometimes we write where (and this is perfectly well-defined because finite fields are unique). The number of elements is .
5) . Let be a finite field and . The set of all matrices with determinant with coefficients in under multiplication will form this group. This is the "special linear group". Sometimes we write where . This group is normal subgroup of . In fact, the mapping (here is the multiplicative group of the finite finite) is a group homomorphism. The kernel is exactly thus we see by fundamental homomorphism theorem that is normal subgroup of and furthermore . Thus, which means . And so we have .
6) . Or as we seen . This where is the group center. A property of this "projective general linear group" is that is is simple. Though, I am NOT sure if it simple for any and any . I do know it is simple almost always, but I just do not know if there are any exceptions to this rule. And I do not know how many elements it has, I never studied it.
7) . The symettries of a regular polygon of vertices. This group has size . It is always a subgroup of . It has size . And it is always non-abelian when .