I know you've been waiting for this:
a nonabelian group that is easy to understand.
The dihedral group Dn is the rotations and reflections of a regular n-gon.
Picture an n-gon with thickness, similar to a coin with edges.
In fact dihedral comes from the latin, meaning two surfaces, like heads and tails.
Assign numbers to the edges of the coin, so you know what is going on.
Place the coin on the table with edge #1 facing front.
Rotate the coin one notch clockwise, then turn it over.
Edge #1 has been shifted to the left,
then flipped over to the right.
Edge #2 is directly in front of you.
Now reverse the operations;
flip the coin and then turn it clockwise.
Edge #1 still faces front after the flip, then shifts left.
This is not where it was before,
hence the group is nonabelian.
The subgroup of pure rotations is Zn,
and it has index 2, hence it is normal in the dihedral group.
All other group elements are involutions,
except for the identity element (leave the coin alone).
For n = 1, the dihedral group is Z2,
like a stick standing up, flipping end over end.
For n = 2,
the coin has two edges,
a front edge and a back edge, and some thickness,
so we can flip it over.
The group consists of vertical flips cross horizontal flips.
These are independent of each other,
hence the group is Z2Z2.
This is not a cyclic group, not the same as Z4,
but it is abelian.
When the coin has 3 or more edges, it looks like a traditional n-gon,
and the group is non-abelian.
The general linear group of order n
is the set of nonsingular n×n matrices under multiplication.
The special linear group is the kernel of the homomorphism implemented by the determinant,
namely the n×n matrices with determinant 1.
These groups are written GLn(R) and SLn(R),
for n×n matrices over a ring R.
