Groups
Why
We further drop conditions on the structure of
the binary operations, and study only the
algebraic structure of addition over the
integers.
Definition
A group is an algebra for which is associative, has an identity element
in , and has inverse elements.
A group is a commutative
group (or abelian
group) if is commutative.
The number of elements is called the
size (or
order) of the group.
A group is a finite
group if is a finite set.
Additive groups
Suppose that is ring.
Then is a commutative group.
Conversely, suppose is a commutative
group.
Define multiplication on by
for all .
Then is a ring, called the
zero ring of .
For this reason, it is customary to write
for the operation when handling
commutative groups.
Group Operations
Along with the group operation, we call the
function which maps an element to its inverse
element the group
operations.