Elementary Properties of Groups A group is a non-empty set G with a binary operation (multiplication or addition) that satisfies specific properties....
A group is a non-empty set G with a binary operation (multiplication or addition) that satisfies specific properties.
Associative property:
For any elements a, b, and c in G, we have:
(ab)c = a(bc).
Identity element:
There exists an element e in G such that, for any a in G, we have:
ea = a and ae = a.
Inverse element:
For each a in G, there exists an element b in G such that:
ab = e and ba = e,
where e is the identity element.
Symmetry property:
If G is a group and g is an element of G, then the element g conjugated with itself under the binary operation is equal to the identity element e.
Dihedral groups:
A dihedral group is a group that is the symmetry group of a regular polygon.
Here are some examples of dihedral groups:
The Dihedral group D4 consists of the symmetries of a square, such as rotations and reflections.
The Dihedral group D6 consists of the symmetries of a regular hexagon, such as rotations and reflections.
The Dihedral group D8 consists of the symmetries of a regular octagon, such as rotations and reflections.
These are just a few examples of dihedral groups. There are infinitely many other dihedral groups, depending on the number of sides of the regular polygon