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Test Corrections

Students are allowed to correct the in-class portion of the test for a chance to earn back half of the lost points. Corrections must be submitted on separate sheets and are due on Tuesday, November 26, at 4:45 PM. Additionally, we'll delve into the First Isomorphism Theorem and related concepts of homomorphisms, including proofs and important terminology. Students should also prepare for discussion questions about group theory concepts, including cyclic and abelian groups. Complete exercises 20-27 on pages 220-221 for practice.

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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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