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Unformatted text preview: Introduction to Groups Keith E. Emmert Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Introduction to Groups Keith E. Emmert Tarleton State University January 19, 2010 Introduction to Groups Keith E. Emmert Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Overview Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Introduction to Groups Keith E. Emmert Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Binary Operations Definition 1 A binary operation ∗ on a set G is a function ∗ : G × G → G . ◮ For any a , b ∈ G we write a ∗ b for ∗ ( a , b ) . ◮ A binary operation ∗ on a set G is associative if for all a , b , c ∈ G we have a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c . ◮ If ∗ is a binary operation on a set G we say elements a and b of G commute if a ∗ b = b ∗ a . We say ∗ or G is commutative if for all a , b ∈ G , a ∗ b = b ∗ a . ◮ Let G be a set with a binary operation ∗ . Suppose H ⊂ G . If for all a , b ∈ H , a ∗ b ∈ H then we say that H is closed under ∗ . Introduction to Groups Keith E. Emmert Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Binary Operations Example 2 ◮ + (usual addition) is a commutative binary operation on Z , Q , R , C and many other sets. ◮ The usual multiplication is a commutative binary operation on Z , Q , R , C and many other sets but not on the set of n × n matrices. ◮ The usual subtraction is a noncommutative binary operation on the set of m × n matrices, Z , Q , R , C and many other sets but fails to be a binary operation on the set of positive integers, Z + . Introduction to Groups Keith E. Emmert Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Group Definition 3 A group is an ordered pair ( G , ∗ ) where G is a set and ∗ is a binary operation on G satisfying the following axioms: 1. Associative: ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) for all a , b , c ∈ G . 2. Identity: there exists an element e ∈ G such that for all a ∈ G we have a ∗ e = e ∗ a = a . 3. Inverse: for each a ∈ G there is an element a 1 ∈ G , called an inverse of a such that a ∗ a 1 = a 1 ∗ a = e . The group ( G , ∗ ) is called abelian or commutative if a ∗ b = b ∗ a for all a , b ∈ G ....
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 Spring '08
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 Dihedral group, Keith E. Emmert

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