Chapter 7:
Cosets and Lagrange’s Theorem
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Properties of cosets
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Lagrange’s theorem and cosequences
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An application of cosets to permutation groups
Definition: Coset of H in G
Let G be a group and H be a subgroup of G. For any a  G, the set
aH  {ah : h  H } is called the left coset of H in G containing a. While
Ha  {ha : h  G} is called the right coset of H in G containing a
More examples
Notes
Proof
Proof; continue
Example;
Proof:
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Definition:
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Corollary 1:
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Corollary 2:
Corollary 3:
Corollary 4:
Example: Fermat’s Little Theorem
Is p  2
Solution:
257
 1 a prime number?
Note that stab G (i) is a subgroup of G.
While stab D4 ( p)  {R0 , D}.
Proof:
Theorem 7.3