To test whether a set along with an operation on the members of the set forms a group, we must test the group properties in turn. These are
G1: Closure – if
then
where
is the group operation.
G2: Associativity – if
then
G3: Identity – every group has an identity element
satisfying
for all
G4: Inverses - ifthen
where
and
Only one of the group axioms needs not to be satisified for a set to fail to form a group.
Example: The set
is a group. We test that the group properties are satisfied one by one.
G1: We can construct the group table.
|
{}times{}-5 |
1 |
2 |
3 |
4 |
|
1 |
1 |
2 |
3 |
4 |
|
2 |
2 |
4 |
1 |
3 |
|
3 |
3 |
1 |
4 |
2 |
|
4 |
4 |
3 |
2 |
1 |
All the highlighted elements are in the set so the closure axiom is satisfied.
G2: Associativity is a property of multiplication, so G2 is satisfied.
G3: The identity element is 1 since
and
and
so G3 is satisfied.
G4: We can read off the inverse of each element from the group table. To find the inverse of an element, find that element in the left margin. Go along in the corresponding row to the identity elemt then up to the top margin. The elemt in that cell is the inverse.
The inverse of 1 is 1
The inverse of 2 is 3
the inverse of 3 is 2
The inverse of 4 is 4.
The set of matrices of the form
is not a group under matrix multiplication. Axiom G1 is not satisfied, since
and this is not of the required form.