StudyGuideCosets

StudyGuideCosets - g G , then g | G | = e . When G = U ( p...

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Cosets and Lagrange’s Theorem Study Guide Outline 1. Cosets Let G be a group, and let H G . Given an element g G , the left coset of H containing g is the set gH = { gh : h H } Each left coset of H has the same size as H , and the left cosets of H form a partition of G . Similarly, the right coset of H containing g is the set Hg = { hg : h H } Each right coset of H has the same size as H , and the right cosets of H also form a partition of G . The number of cosets of H in G is called the index of H in G , and is denoted | G : H | . Since the size of every coset is | H | , we have | G : H | = | G | / | H | Problems: 23, 24, 25 2. Lagrange’s Theorem Lagrange’s theorem states that: If G is a finite group, and H G , then | H | is a factor of | G | . This follows from the fact that the cosets of H form a partition of G , and all have the same size as H . Applying this theorem to the case where H = h g i , we get If G is a finite group, and g G , then | g | is a factor of | G | . It follows easily that Every group of prime order is cyclic. Problems: 26, 28 3. Consequences in Number Theory If G is a group and g G , then the order of g must be a factor of | G | . Equivalently, If G is a finite group, and
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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 p-1 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 orbit-stabilizer theorem . It can also be written as follows stab G ( k ) orb G ( k ) = | G | Problems: 29, 30...
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StudyGuideCosets - g G , then g | G | = e . When G = U ( p...

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