Lecture 4 - 4 Examples of groups Consider the set a,b and...

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4. Examples of groups Consider the set { a, b } and define a multiplication rule by aa = a ab = b ba = b bb = a Here a plays the role of the identity. a and b are their own inverses. It is not hard to check that associativity holds and that we therefore get a group. To see some more examples of groups, it is first useful to prove a general result about associativity. Lemma 4.1. Let f : A -→ B , g : B -→ C , h : C -→ D be three functions. Then h ( g f ) = ( h g ) f. Proof. Both the LHS and RHS are functions from A -→ D . To prove that two such functions are equal, it suffices to prove that they give the same value, when applied to any element a A . ( h ( g f ))( a ) = h (( g f )( a )) = h ( g ( f ( a ))) . Similarly (( h g ) f ))( a ) = ( h g )( f ( a )) = h ( g ( f ( a ))) . The set { I, R, R 2 , F 1 , F 2 , F 3 } is a group, where the multiplication rule is composition of symmetries. Any symmetry, can be interpreted as a function R 2 -→ R 2 , and composition of symmetries is just composition of functions. Thus this rule of multiplication is associative by (4.1).
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