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 (10/22) at 4:45.

2. Some Exercises on Isomorphisms Etc. (10/18/13) On yes-no questions, A is yes and B is no. G and G’ are non-trivial abstract groups. “OP” means operation-preserving. Whatever your response, think of justification: Is every OP function from G to G’ an isomorphism? Is every 1-to-1 and onto function from G to G’ an isomorphism? Is every automorphism an isomorphism? Is every isomorphism an automorphism? Is every automorphism of G a permutation of G? Is every permutation of G an automorphism of G?

3. Exercises Continued Whatever your response, think of justification: Is  R 90  in D 4 isomorphic to U(8)? Is Z 4 isomorphic to U(10)? Is U(125) isomorphic to U(184)? Is R (under +) isomorphic to R + (under times)? Is Q (under +) isomorphic to Q + (under times)? Is 7Z isomorphic to 8Z? Does S 6 contain subgroups isomorphic to S 4 ? (If so, how many do you think?) Does A 6 contain subgroups isomorphic to D 4 ?

4. Exercises Continued Whatever your response, think of justification: In Z 10, if  takes 1 to 6, is  an automorphism? In Z 10, if  takes 1 to 9, is  an automorphism? In Z, if  takes 1 to 6, is  an automorphism? In Z, if  takes 1 to 6, is  an isomorphism? In C, is the “conjugate mapping”  which takes x + i y to x – i y an automorphism? In D 4, if  takes R 90 to R 180, is  an automorphism? In D 4, if  takes R 90 to R 270, D to D’, and H to V, do you think  is an automorphism?

5. Assignment for Monday Work on Test Corrections! We will set Monday aside for an optional question and answer period. At that time we can discuss issues arising in the test corrections and also issues from the take-home part. Please look at the answer key on that part before coming to that class. Keep working on Chapter 6.