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Test Corrections

Test Corrections. You may correct the in-class portion of your test . You will get back 1/2 of the points you lost if you submit correct answers. This work is to be done on your own (or in consultation with me only). Skidmore Honor Code!

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Test Corrections

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  1. Test Corrections • You may correct the in-class portion of your test. You will get back 1/2 of the points you lost if you submit correct answers. • This work is to be done on your own (or in consultation with me only). Skidmore Honor Code! • Corrections should be done on separate sheets, NOT on the original test. Hand both things in, NOT stapled together. • Due on Tuesday (11/26) at 4:45.

  2. More on Homomorphisms (11/22) • “First Isomorphism Theorem”. If  is an homomorphism from G to G’, then the image of , (G) is isomorphic to G / Ker  . • Proof. We’ve pretty much already done it. Define  from G/ Ker  to (G) by (a Ker) = (a). We showed last time that  is well-defined, one-to-one, and onto. Hence we need only establish OP. But by the definition of coset multiplication, we have ((a Ker )(bKer )) = (a bKer ) = (a b) = (a)(b) = (aKer ) (b Ker )) .

  3. Some Questions • If  is the “natural” modular homomorphism from Z to Zn , What factor group of Z is Zn isomorphic to? • What “well-known” group is GL(2, R) / SL(2, R) isomorphic to? Why? • Terminology: If  is an homomorphism from G to G’, then (G) is called a homomorphic image of G. • If G is abelian, is (G) necessarily abelian? • If (G) is abelian, is G necessarily abelian? • If G is cyclic, is (G) necessarily cyclic? • If (G) is cyclic, is G necessarily cyclic?

  4. Assignment for Monday • Work on test corrections. • On page 220-221 do Exercises 20-27.

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