## Properties of Permutations

Every permutation  can be written as a cycle or a product of disjoint cycles. This follows by considering the effect of a sequence of permutations on each member of the set Each element i will end up after a sequence of permutations in some other position meaning that somewhere is the simplified result, the sequence …, i, j, … must occur. There cannot be somewhere else a sequence of the form …, j, k, … or …, k, j, …since if this occurred, we would be able to simplify it.

Disjoint cycles commute. If and and the cycles have no element in common, then Since and are disjoint we may write where the 's are the symbols in neither or Suppose is permuted by then since fixes all the terms.

Similarly so for each permuted by A similar argument is made for terms permuted by and of course, both and leave the terms fixed. Hence The order of a permutation of disjoint cycles is the least common multiple of the lengths of the cycles. To prove this, observe that a cycle of length has order Suppose then that and are disjoint cycles of length and and let be the least common multiple of and  and are the identity permutation, and since and commute Then the order of must divide Suppose the order of is then so but if and are disjoint, then so are and so both must be equal to and both and divide hence that divides also and The argument can be extended to any number of cycles in an obvious way.

Every permutation can be written in cycle form as a 2 cycle or a product of 2 cycles.

Note that and Permutations are odd or even. A permutation is even if it can be written as an even number of 2 cycles and odd if it can be written as an odd number of 2 cycles. Since every permutation can be written as a product of two cycles, the result follows.

The set of even permutations of forms a subgroup of To prove it, write each element as a product of two cycles.

Define then the kernel of is the set of all even permutations.

The product of even permutations is even, as is the identity, and the inverse of an even permutation, so that the set of all even permutations is a subgroup. 