Group (mathematics) - Finite Groups

Finite Groups

A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups S_N, the groups of permutations of N letters. For example, the symmetric group on 3 letters S₃ is the group consisting of all possible orderings of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S_N for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S₃ can also be interpreted as the group of symmetries of an equilateral triangle.

The order of an element a in a group G is the least positive integer n such that a n = e, where a n represents

i.e. application of the operation • to n copies of a. (If • represents multiplication, then an corresponds to the nth power of a.) In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any finite subgroup H divides the order of G. The Sylow theorems give a partial converse.

The dihedral group (discussed above) is a finite group of order 8. The order of r₁ is 4, as is the order of the subgroup R it generates (see above). The order of the reflection elements f_v etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem. The groups F_p× above have order p − 1.