Category Archives: free group
diagram chasing the wreath
Having an action between two groups means a map that comply
Then one can assemble a new operation on to construct the semidirect product . The group obtained is by operating
Let be a set and the set of all maps . If we have an action then, we also can give action via
Then we define
the so called permutational wreath product.
This ultra-algebraic construction allow to give a proof of two pillars theorems in group theory: Nielsen – Schreier and Kurosh.
The proof becomes functorial due the properties of this wreath product.
The following diagram is to be exploited
Let be the rank two free group and be a subgroup.
Observe that , then .
Clearly , because it is not difficult to convince oneself that consists on words of even length and implies .
Technically, that is attending to the Schreier’s recipe, having as a set of transversals and being the free generators for .
Set and take , then we get
So according to Schreier’s language the set in our case, is
Hence are the free generator for .
Note that this three word are the first three length-two-words in the alphabetical order, start by and continuing to
that the presentation given by
determines the group ?
Similar question for
for the group .
Other similar problems but less “difficult” are:
- it is
- is , the rank two free group
- dare altri venti esempi
that is, writing without spaces between the words… is it a fashion?