Groups

Let be a set together with a map .

Associativity: for all
Unit element: There exists such that for all
Inverse elements: For all there exists such that
Commutativity: for all

Let be a set together with a map . Then is called a...

Semigroup if it is associative
Monoid if it is associative and has a unit element
Group if it is associative, has a unit element, and has inverse elements
Abelian group if it is a commutative group

As associativity holds, we can simply write without parentheses. We also commonly define for all , and if a unit element exists. Finally, we often write as .

Let be a group. Then the following holds:

1. The unit element is unique.
2. The inverse of any is unique (denoted by ).
3. for all .
4. If or , then .

Proof:

1. If and are unit elements, then by the properties of unit elements.
2. If and are inverses of , then .
3. We know , so is the inverse of .
4. If , then , and analogous if .

Let be a set with a map that satisfies

1. is associative
2. has a left unit such that for all .
3. has left inverses; for all there exists a such that .

Then, is a group.

Proof:
We first show that if is the left inverse of , it is also the right inverse of . We have , where the last step follows by applying the left inverse of on both sides. Hence, inverses exist. We also get , so is indeed also a left unit.

1. is an abelian group.
2. If is a field, is an abelian group.
3. If is a field and is a vector space over , is an abelian group.
4. If is a field, is an abelian group.
5. and are abelian groups.
6. , , and are commutative monoids, but not groups.