Group
A group is a monoid that has an underlying set for which each element can be paired with another element that, when combined together with the monoid's monoid operation, gives you the identity element. 1
Types
- Cyclic group is a monoid for which there is one generator which can be applied successively that produces all of the elements in the set. 1
- Dihedral groups is a group of rotations and reflections of any regular polygon. 2
Isomorphism
Two groups are isomorphic if, given an function f from one group to the other and any two elements a and b from the first group, f(a ∘ b) is the same as f(a) ∘ f(b). 2
Presentation
The presentation of a group is the set of generators that defines a given group.
Examples:
| Group | # of morphisms | Concrete example |
|---|---|---|
| Z3 | 3 | For a regular triangle: identity (0 degrees), 120 degrees, and 240 degrees. |
| Klein four-group | 4 | For a non-square rectangle: identity, horizontal flip, vertical flip, and the horizontal and vertical flip (180 degrees). |
Product
A group product is the combination of two groups by creating a new group containing an underlying set that is the cartesian product of the original two groups and the operations of both groups.
Examples:
- The group product of two Z2 groups is known as the Klein four-group and is a non-cyclic Abelian group. 2
A group product results in an Abelian group if the groups used in the product are also Abelian. All Abelian groups are either cyclic or products of cyclic groups, which means that the only way to make non-cyclic Abelian groups is by doing a product of cyclic groups. The Klein four-group is an example of a non-cyclic Abelian group. 2