One interesting point about the three subsets. If you take any two elements, a and b out of one of the subsets (both from the same subset) then their difference, a - b, will be a multiple of three. Their difference will be in the subgroup that originally generated them.
Another example occurs in the Atayun-HOOT! group.
| • | A | B | R | L |
|---|---|---|---|---|
| A | A | B | R | L |
| B | B | A | L | R |
| R | R | L | B | A |
| L | L | R | A | B |
A and B form a subgroup as we have already seen. R and L form a coset of that subgroup. The property mentioned in the multiple-of-three example holds here. If any member of {R, L} is multiplied by the inverse of any member of {R, L} the result is in the {A, B} subgroup.
Let's look at another example, the group of symmetry movements of the equilateral triangle. Its elements are, in the notation we gave in chapter 5, e, a, b, X, Y, Z.
Let's look again at the Cayley table for this group:
| • | e | a | b | X | Y | Z |
| e | e | a | b | X | Y | Z |
| a | a | b | e | Y | Z | X |
| b | b | e | a | Z | X | Y |
| X | X | Z | Y | e | b | a |
| Y | Y | X | Z | a | e | b |
| Z | Z | Y | X | b | a | e |
A glance at the north-west corner of the Caley table for this group shows that the subset {e, a, b} is closed under the group operation followed by. Since it is a finite subset this suffices for it to be a subgroup. A coset of the subgroup is {X, Y, Z}. Note again that any member of this coset multiplied by the inverse of any member gives an element of the subgroup.
If you look at the table again you will notice that the subset
{e, X} is also closed under the group operation. As
it is finite we know that it also is a subgroup of the main group.
Cosets of this group are {Y, b} and {Z, a}.
These cosets obey the property that if a and b are in
the coset then a • b-1 is in the subgroup. But
what about a-1 • b? Let's try the coset
{Y, b} Note that
which is not in the subgroup. However the cosets {Y, a) and {Z, b} obey the property that if u and v are in the coset then u-1 • v is in the subgroup. Thus the cosets with the u • v-1 -is-in-the-subgroup property and the cosets with the u-1 • v-is-in-the-subgroup property are different cosets.
The cosets that have the u • v-1 -is-in-the-subgroup property are called right cosets and the cosets that have the u-1 • v-is-in-the-subgroup property are called left cosets.
[Exercise: Find the left and right cosets for the subgroup {Y, e}. Find the left and right cosets for the subgroup {Z, e}.]
Note that for the subgroup {e, a, b}, that we previously looked at, {X, Y, Z} is both a right coset and a left coset.
[Exercise: Verify the last statement.]
Subgroups whose right cosets are also left cosets are very important in group theory. They are called normal subgroups.
Again let's take the subgroup {e, X} and mulitply each member of the subgroup from the right by Y.
We get {Y, b} which was one of the right cosets of the subgroup {e, X} that we found previously. The other element of the right coset is b. Let's multiply each member of the subset {e, X} by b and see what happens
We end up with the same coset again. This, as it turns out, always works and is the usual way of defining a right coset of a subgroup.
Definition: If H is a subgoup of a group G then for any element g of the group the set of products of the form h • g where h is in H is a right coset of H denoted by the symbol Hg. The set of all products of the form g • h where h is in H is a left coset of the subgroup H denoted by the symbol gH.
In fact if we take any subset U of a group G (which can be a subgroup or not) we can multiply every element in it from the right by some group element v to get another subset V of G. If the subset H happens to be a subgroup of G then the subset it is transformed into is called a right coset of the subgroup H.
In set-builder notation we can define Uv as
When U is a subgroup this is a right coset of U.
The operation of "multiplication on the right by a group element" which transforms subsets into subsets (and subgroups into right cosets) will also transform any right coset of a subgroup H into a right coset of H (This could be a different right coset than we started with or the same one). This follows from the definition since (Hu)v is all elements of the form (h • u) • v which by the associative property is the same as the set of elements of the form h • (u • v) which is none other than H(u • v).
Of course similar stuff is true of left cosets. Just hold your brain up to a mirror while doing proofs of right-coset properties.
Can we recover our original property of right and left cosets from this definition?
THEOREM: If x and y are in the same right coset of a subgroup H then x • y-1 is in H.
Proof: If x and y are in the same right coset of H then there are h1 and h2 in H and some element a of G such that
By one of our housekeeping theorems the inverse of y is (h2 • a)-1 = a-1 • h2-1. Thus
By closure of the subgroup h1 • h2-1 is in H and our original property is preserved by the new definition.
THEOREM: If x • y-1 is in subgroup H of group G then x and y are in the same right coset of H.
This is the converse of the previous theorem. It also establishes two equivalent definitions of "coset."
Proof: If x • y-1 is in H then
for some h1 in H. It then follows that
and therefore x is in Hy. Also since e the identity element of G is in H it follows that
is also in Hy.
Thus x and y are both in the same coset of H, namely Hy.
[Exercise: State and prove the parallel theorems concerning left cosets.]
Lagrange's Theorem
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© 1998 by Arfur Dogfrey