Group (mathematics)/Catalogs

The mathematical group concept represents a rather simple and natural generalization of common phenomena, so examples of groups are easily found, from all areas of mathematics.

Different classes of groups

Three different classes of groups are commonly studied:

• Finite discrete groups
• Infinite discrete groups
• Continuous groups

Examples of finite discrete groups

Illustration of the cyclic group of order 4. Let r1 be the act of turning the knob 1 step clockwise. Let r2 be the act of turning the knob 2 steps clockwise. Let r3 be the act of turning the know 3 steps clockwise. Finally, let r0 be the act of just doing nothing. It's easy to see the following: Doing r1 and then r1 again gives the same result as doing r2. Doing r1 and then r2 gives the same result as doing r3. Doing r1 and then r3 gives the same result as doing nothing, i.e. r0. ... These results can be summarized in the following table: * r0 r1 r2 r3 r0 r0 r1 r2 r3 r1 r1 r2 r3 r0 r2 r2 r3 r0 r1 r3 r3 r0 r1 r2

1. The trivial group consisting of just one element.
2. The group of order two, which f.i. can be represented by addition modulo 2 or the set {-1, 1} under multiplication.
3. The group of order three.
4. The cyclic group of order 4, which can be represented by addition modulo 4.
5. The noncyclic group of order 4, known as the "Klein viergruppe". A simple physical model of this group is two separate on-off switches.

Many examples of groups come from considering some object and a set of bijective functions from the object to itself, which preserve some structure that this object has.

• Topological groups:
  • Matrix groups; e.g the general linear group, the special linear group, the projective linear group and the Platonic groups
  • Abelian varieties.
    • Elliptic curves.
    • Abelian surfaces.
• Finite groups.
  • Cyclic groups.
  • Symmetric groups and alternating groups.
  • Dihedral groups.
  • Group of quarternions.
• Galois groups.
• Fundamental groups.