A color-coded example of non-trivial abelian, non-abelian, and normal subgroups, quotient groups and cosets.

This image shows the multiplication table for the permutation group
S4, and is helpful for visualizing various aspects of groups. This
group consists of all the permutations possible for a sequence
of four numbers, and has 24 (= 4!) elements. For example, the
element **r1** rotates the sequence one place to the left. That is,
if the original sequence looks like (1234) then **r1** maps
(1234) -> (2341). The mirror element **m0** flips the order, i.e.
(1234) -> (4321). The combination of **m0*r1** first rotates, then
mirrors, so the map is (1234) -> (1432), which is defined in the
Key as element **m1**, so **m0*r1** = **m1**.

There are many things to notice here.

- S4 is not abelian. For example,
**r1*d3**=**c0**, but**d3*r1**=**d0**. Geometrically, if you fold the table about its diagonals the elements do not match. - The set
**{e,r1,r2,r3}**is an abelian subgroup Y of S4 that has 4 elements and is marked in yellow. If you start in yellow and interact with yellow you stay in yellow (what was that saying about Vegas?). - The set M =
**{e,r1,r2,r3,m0,m1,m2,m3}**(generated from products of the mirror and rotation elements**{r1,m0}**) is also a subgroup of S4. M has eight elements, is non-abelian, and contains the subgroup Y. That is, if you interact purple with yellow you get purple or yellow. - The left cosets (L_h) of the subgroup Y are defined as the set of
all elements h*Y for a given element h in S4. h is a member of the coset L_h
because e is in Y and h = h*e. The possible products
**(h*y1)*y2 = h*(y1*y2) = h*y3**are still some combination of h*y_i; so in this case each coset is simply a four-element set formed by**{h*e,h*r1,h*r2,h*r3}**. For example, with h=c1 we get the coset**{c1,c2,c3,c0}**. The six left cosets form blocks of color (yellow, purple, green, orange, blue, and white) at the left of the diagram. - If we had similar color blocks on the right, then the blocks themselves
would act as a group, the quotient group S4/Y. But this does not work because
the right cosets differ from the left cosets in this case. For example, a0*r1 = a1 happily lies
in the same green left coset as a0, but in the right coset of a0, r1*a0 = d1 is white. Thus,
the right and left cosets of S4/Y differ.
To make the cosets act like a group we need

**(g^-1)*Y*g**to be in Y for all g. A subgroup satisfying this requirement is called a*normal*subgroup. We could can only make a quotient group G/N if the subgroup N is normal. In our case,**a1*b3=e**, so**b3=(a1^-1)**. We would need**b3*y*a1**to be in Y for all y in Y if Y were normal. But**b3*(r1*a1)**=**b3*d2**=**c2**, and this is not in Y (orange, not yellow). - It is interesting that the subgroup M (yellow plus purple) produces nice color blocs everywhere on the left. Meaning...
- Y is a normal subgroup of M. Elements of M are either yellow (y) or purple (m).
(y1^-1)*y2*y1 is the product of three elements in Y and remains in Y. From the
table, all the m in {m0,m1,m2,m3} are their own inverses:
**(m^-1)**=**m**. Hence,**(m^-1)*y*m**=**m*(y*m)**. But all y*m produce another m, and all m*m give a y. Therefore M is normal, and M/Y is a group of order 2 (the four full color blocks in the upper left corner). - Now if instead you picked the 8-element set M (all yellow+purple) as the subgroup and colored the cosets that way, then yellow+purple = e*M would be one left coset, green+orange=a0*M a second one, and blue+white=b0*M a third. As before, the right cosets are different from the left ones, and M is not normal in S4.