tsüklivõru

tsüklivõru is a term used in Estonian to describe a cyclic permutation. A cyclic permutation is a permutation of the elements of a set that moves some elements in a cycle while keeping others fixed. For instance, if you have the set {1, 2, 3, 4} and a permutation that maps 1 to 2, 2 to 3, and 3 to 1, with 4 remaining in its original position, this is a cyclic permutation. The cycle is (1 2 3), and 4 is a fixed point.

In mathematics, a tsüklivõru can be represented using cycle notation. A cycle like (a b c) signifies that element 'a' is replaced by 'b', 'b' by 'c', and 'c' by 'a'. Any elements not explicitly mentioned in the notation are considered to be fixed. The order of a tsüklivõru is the length of the cycle. A single element cycle, like (a), represents a fixed point and has an order of 1.

The concept of tsüklivõru is fundamental in group theory, particularly in the study of symmetric groups. Any permutation can be decomposed into a product of disjoint cycles. Understanding tsüklivõru is crucial for analyzing the structure and properties of these permutations. This decomposition helps in determining properties such as the parity of a permutation, which indicates whether it can be achieved through an even or odd number of transpositions (swaps of two elements).