Conjugation



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Conjugation

Having described the elements of , we show which of them are in the same conjugacy class, i.e. in the same orbit of the group on the set under the conjugation action (cf. gif). In order to do this, we first note how is obtained from the permutation :

Thus, in terms of cyclic factors of , arises from

by simply applying to the points in the cycles of :

 

(For any mapping we mean by that has to be replaced by and not that is replaced by as it is sometimes understood!) This equation shows that the lengths of the cyclic factors of are the same as those of . It is easy to see that, conversely, for any two elements with the same lengths of cyclic factors there exists a such that . Hence the lengths of the cyclic factors of characterize its conjugacy class. To make this more explicit, we introduce the notion of (proper) partition   of , by which we mean any sequence of natural numbers which satisfy

The are called the parts   of . The fact that is a partition of is abbreviated by

If then there exists an such that for all . We may therefore write

for any such . The minimal with this property will be denoted by and called the length of . The following abbreviation is useful in the case when several nonzero parts of are equal, say parts are equal to :

If , then is usually omitted, e.g. For the ordered lengths , of the cyclic factors of in cycle notation form a uniquely determined proper partition

which we call the cycle partition of . The corresponding -tuple

consisting of the multiplicities of the parts of length in is called the cycle type of . Correspondingly we call an -tuple a cycle type of if and only if each , and This will be abbreviated by



next up previous contents
Next: Conjugacy Classes Up: Actions Previous: Cycle decomposition



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995