This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: g ∈ G , then g  G  = e . When G = U ( p ), where p is prime, this gives us Fermat’s 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 Euler’s 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...
View
Full Document
 Spring '09
 Algebra, Group Theory, Sets, Equivalence relation, Cyclic group, Coset, Lagrange’s Theorem

Click to edit the document details