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.

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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. 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  R given 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 kernel of a 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 coset a 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 D5 to Z10? What are they exactly? Example. How many homomorphisms are there from D4 to Z7? What are they exactly? Example. How many homomorphisms are there from D4 to Z8? What are they exactly? For Friday, please read Chapter 10 to page 214 and do Exercises 1-5, 9, 14, 15, 16, 19.