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Homomorphisms (11/20). Definition. If G and G ’ are groups, a function from G to G ’ is called a homomorphism if it is operation preserving, i.e., for all a , b in G , ( a b ) = ( a ) ( b ). Example. Every isomorphism is automatically a homomorphism, but not conversely.
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Homomorphisms (11/20) • Definition. If G and G’are groups, a function from G to G’is called a homomorphismif it is operation preserving, i.e., for all a, b in G, (a b) = (a) (b). • Example. Every isomorphism is automatically a homomorphism, but not conversely. • Is : R* R* given by (x) = x2 a homomorphism? Is it an automorphism? • Is : R+ R+given by (x) = x2 a homomorphism? Is it an automorphism? • Is : R Rgiven by (x) = x2 a homomorphism? • Note that for any pair of groups G and G’, there is always at least homomorphism between them. What is it?
Kernels and Images • Definition. The kernelofa homomorphism : G G’, denoted Ker, is {a G| (a) = e’} , where e’ is the identity of G’. • Theorem. Ker is a normal subgroup of G. • Proof. You did it (right?) on the hand-in. • Definition. The image of , denoted (G) is {c G’| (a) = c for some a G}. • Theorem. (G) is a subgroup of G’. • Proof?? • If is an isomorphism from G to G’, then Ker = ??? and (G) = ???
A Handy Theorem and Corollaries • Theorem. If (a) = c, then {x G| (x) = c} is exactly the coseta Ker. • Proof. (How do we show two sets are equal?) • Corollary. If |Ker | = k, then is a k to 1 function. • Corollary. If G is finite and if H G, |(H)| = |H| / |Ker |. • Corollary. If G and G’ are both finite, then|(G)| divides both |G| and |G’|. • Example. How many homomorphisms are there from Z4 to Z3? What are they exactly? • Example. How many homomorphisms are there from Z4 to Z2? What are they exactly? • Example. How many homomorphisms are there from Z4 to Z6? What are they exactly?
More Examples and Assignment • Example. How many homomorphisms are there from D5to Z10? What are they exactly? • Example. How many homomorphisms are there from D4to Z7? What are they exactly? • Example. How many homomorphisms are there from D4to Z8? What are they exactly? • For Friday, please read Chapter 10 to page 214 and do Exercises 1-5, 9, 14, 15, 16, 19.
