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Unformatted text preview: g G , then g  G  = e . When G = U ( p ), where p is prime, this gives us Fermats little theorem : If p is prime and x is not a multiple of p , then x p1 1 mod p . More generally, if G = U ( n ) this gives us Eulers theorem : If x and n are relatively prime, then x  U ( n )  1 mod n . Problems: 27 4. Orbits and Stabilizers If G S n and k { 1 ,...,n } , then the stabilizer of k in G is following subgroup of G : stab G ( k ) = { g G : g ( k ) = k } . The orbit of k under G is the following subset of { 1 ,...,n } : orb G ( k ) = { g ( k ) : g G } . The stabilizer of k has one coset for each element of the orbit of k . That is, G : stab G ( k ) = orb G ( k ) This is known as the orbitstabilizer theorem . It can also be written as follows stab G ( k ) orb G ( k ) =  G  Problems: 29, 30...
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This document was uploaded on 11/03/2010.
 Spring '09
 Algebra, Sets

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