isomorphism

[enzocreti]

Let be E the parity function S(n) in -1,+1


where S(n) are the permutations.


S(n)/A(n) is the quotient Group where A(n) denotes Ker E


so the quotient Group is given by the right (left cosets)of odd and even permutations?

[Nick]

Yes, as long as n is at least 2, that quotient group has 2 elements: the set of all even pemutations and the set of all odd permutations.
These are the 2 cosets of A(n) in S(n).

[enzocreti]

so it follows that S(n)/A(n) is isomorphic to the Group (-1,+1)?
by the fundamental theorem of isomorphism?

[Nick]

Originally Posted by enzocreti

so it follows that S(n)/A(n) is isomorphic to the Group (-1,+1)?
by the fundamental theorem of isomorphism?

Yes, for all n≥2, that's right.

[enzocreti]

ok thanks