Consider the subgroup of integers divisible by 3. This forms a subgroup of the additive group of integers. Its elements are { . . . -9, -6, -3, 0, 3, 6, 9, . . .}. By adding 1 to any multiple of 3 we get another subset of group of integers, { . . . -8, -5, -2, 1, 4, 7, 10, . . .}. By adding 2 to the elements of the multiples-of-3 subgroup gives us the subset { . . . -7, -4, -1, 2, 5, 8, 11, . . .}. These three subsets exhaust the integers. Any integer will be in one and only one of these subsets.

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**.

**e • Y = Y****X • Y = b**

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

**e • b**= b**X • b**= Y

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 *H***g**. 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 **g***H*.

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 *U***v** 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 transforms 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
(*H***u**)**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 **h**_{1} and **h**_{2} in *H* and some element **a** of *G* such that

By one of our housekeeping theorems the inverse of **y** is **(h**_{2}** • a)**^{-1}** = a**^{-1}** • h**_{2}^{-1}. Thus

By closure of the subgroup **h**_{1}** • h**_{2}^{-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 **h**_{1} in *H*. It then follows that

and therefore **x** is in *H***y**. Also since **e** the identity element of *G* is in *H* it follows that

*
is also in Hy*

**
[ Exercise: State and prove the parallel theorems concerning left cosets.]
**

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© 1998 1999 by Arfur Dogfrey