Permutation Groups

The operation of a group is closed on a group's own set of elements. A product of two elements of a group gives another element of the group. Since the cancellation laws apply in groups to products of elements, (i.e. ab = ac implies b = c) we can represent an element in a group as a permutation of the groups elements only. We then obtain all the information of the operation of multiplication by an element to the permutation of the groups elements under "action" by that element. The cancellation laws allow there to be a unique answer for each element's "permutation" the result of the product with any other element(s). In this manner, a group is mapped onto itself in a 1-1 manner (bijection) by each element. Simply by noting the effect multiplication by any element has, we find not only a definition for an element, but the effect of the group's operation by any element upon the others.

Whilst mapping the set of elements {a,b,c,....} to a permuted set, it becomes evident that a series of composite 'Transpositions' (or the swapping of two elements at a time) preserves the 1-1 map and may suffice for any permutation of the elements. Permutations may be written as a series of disjoint cycles of arbitrary lengths. A cycle of length 'n' represents an order 'n' element. (1,2,3,4) is order four and maps 1 to 2, 2 to 3, 3 to 4 and 4 to 1. In two cycles, (transpositions) this factors as; (3,4)(2,4)(1,4). This notation acts onto the right, much as one would write function composition. f(g(x)). It reads like, (1 goes to 4 then 2, 2 goes to 4 then 3, 3 goes to 4, 4 goes to 1).

The trivial cycles (4,4), (5,5,5) etc are represented simply by the letter 'e'. Any group G of 'n' elements whether abelian or not, may be considered a subgroup of the permutations on a set of n elements. All such permutations in n symbols form a group Sn, named the symmetric group of degree n. This group Sn has n! elements (5!=5 x 4 x 3 x 2 x 1).

The largest subgroup of Sn is An, the alternating group of degree n, with order n!/2. This is a group composed of all elements in Sn that factor into an even number of transpositions. This may be quite complicated, do not confuse it with even cycle length. some elements have cycle notations like (2,3,4)(2,6,4)(5,7). It is not true that in all cases ab=ba, (commutativity). We should note that disjoint cycles always commute. Element's inverses may be found by reversing the order of their transpositions or "reversing". An inverse to a 'n' cycle is found by reversing it. I.e (1,2,3)(3,2,1)=e. So (1,2,3) has as its inverse (3,2,1)

Any group may be so represented by permutations, without exception.

It is common practice for Abelian groups where the operation is commutative to use the + sign for the operation so a+b = c etc, whereas for non-abelian (non commutative operation) groups multiplication is used so, a*b = c often dropping the * to simply ab = c.

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