Group
A mathematical group is a set of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.
Group is a college-level concept that would be first encountered in an abstract algebra course covering group theory.
Examples
| Abelian Group: |
An Abelian group is a group for which the binary operation is commutative. |
| Cyclic Group: |
A cyclic graph is an (always Abelian) abstract group generated by a single element. |
| Dihedral Group: |
The dihedral group of order n>i is the symmetry group for a regular polygon with n sides. |
| Finite Group: |
A finite group is a group with a finite number of elements. |
| Simple Group: |
A simple group is a mathematical group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. |
| Symmetric Group: |
A symmetric group is a group of all permutations of a given set. |
Prerequisites
| Congruence: |
A congruence is an equation in modular arithmetic, i.e., one in which only the remainders relative to some base, known as the "modulus," are significant. |
| Matrix: |
A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix is an extremely important concept in linear algebra. |
| Permutation: |
In combinatorics, a permutation is a rearrangement of the elements in an ordered list S into a one-to-one correspondence with S itself. Combinatorics studies the number of possible ways of doing this under various conditions. |
Classroom Articles on Group Theory
Classroom Articles on Abstract Algebra (Up to College Level)
