Algebra

Final Exam

Winter 2005

Hunsicker

Definitions (5 points each)

1) Operation on a set

2) The group S_n

3) Group isomorphism

4) Equivalence relation

5) Center of a group

6) Prime ideal

7) Field

8) Zero divisor

Theorems to state (5 points each)

9) Cayley’s Theorem

10) Lagrange’s Theorem

11) The Fundamental Homomorphism Theorem

12) Sylow’s Theorem

Theorems to prove (10 points each): Prove two of the first three theorems above. Clearly indicate which ones you are answering, and cross out any aborted attempts at other proofs.

True/False (2 points each)

T F 13) No operation can have more than one identity element.

T F 14) In any ring, for any elements a and b, -(ab)=(-a)b.

T F 15) An operation must have inverses to have a cancellation law.

T F 16) A cycle with an even number of elements is an even permutation.

T F 17) Conjugacy classes in a group are all the same size.

T F 18) Any subgroup of an cyclic group is cyclic.

T F 19) If h=f(g) where f is a group homomorphism then ord(g) divides ord(h).

T F 20) The composition of injective functions is injective.

T F 21) If aⁿ = e then n=ord(a).

T F 22) The direct product of two cyclic groups is cyclic.