isomorphism

enzocreti · 2019-10-02, 11:57

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 · 2019-10-02, 12:07

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 · 2019-10-02, 12:19

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

Nick · 2019-10-02, 14:38

Quote:

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 · 2019-10-03, 06:30

ok thanks

2019-10-02, 11:57	#1
enzocreti Mar 2018 2·5·53 Posts	isomorphism 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?

2019-10-02, 12:19	#3
enzocreti Mar 2018 2·5·53 Posts	isomorphism so it follows that S(n)/A(n) is isomorphic to the Group (-1,+1)? by the fundamental theorem of isomorphism?

2019-10-03, 06:30	#5
enzocreti Mar 2018 1000010010₂ Posts	isomorphism ok thanks

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2019-10-02, 12:07	#2
Nick Dec 2012 The Netherlands 6A6₁₆ Posts	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).