Math 619 Final Exam - December 11, 2012
1. Short Answer- no work need be shown. (40 points)
a. Give an example of a UFD that is not a PID.
b. Find a generator for the ideal (4 + 10i, 58) in Z[i].
c. Give an example of a ring which is nitely generated with
Math 619 Final Exam SOLUTIONS - December 11, 2012
1. Short Answer- no work need be shown. (40 points)
a. Z[x]
b. -4+10i
c. R = F [x1 , x2 , . . . , ] is a nitely generated left R module but the ideal (x1 , x2 , . . . ) is
not nitely generated.
d. All are
Math 619 Midterm Exam #2- November 1, 2012
1. Short Answer- no work need be shown. (30 points)
a. Give generators for a Sylow 3-subgroup of S9 .
b. Dene a nilpotent group and nilpotence class.
c. Give an example of a group that is solvable but not nilpote
Math 619 Midterm Exam #1- September 27, 2012
1. Short Answer- no work need be shown. (30 points)
a. Dene what it means for a group action to be faithful.
b. Determine the commutator subgroup of D8 and S4 .
c. State the second (aka diamond) isomorphism the
Math 619 Midterm Exam #1 SOLUTIONS
1. a. A group action on A is faithful if g a = a a A implies g = e.
b. (D8 ) = cfw_e, r2 , (S4 ) = A4 .
c. Suppose H, K G with H NG (K ). Then HK G, K HK , H K H and
HK/K H/H K.
=
d. cfw_e, cfw_r2 , cfw_r, r3 , cfw_s, sr