{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

StudyGuideCosets

# StudyGuideCosets - g ∈ G then g | G | = e When G = U p...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the 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 p-1 ≡ 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 orbit-stabilizer theorem . It can also be written as follows ± ± stab G ( k ) ± ± × ± ± orb G ( k ) ± ± = | G | Problems: 29, 30...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online