Review for Final Exam

You are responsible for all of the material we have learned. There will
be twenty true/false questions on the exam (worth 40 points total out of 200)
which test your knowledge of theorems and definitions as well as ten definitions
to state, ten theorems to state and three theorems to prove.

Definitions to state
Divides
Greatest common divisor
Relatively prime
Group
Subgroup
Product of groups
Group isomorphism
Order of a group
Order of a group element
Cyclic group
Partition
Equivalence relation
Equivalence class
Conjugate elements of a group
Coset
Group homomorphism
Kernel of a homomorphism
Image of a homomorphism
Quotient group
Center of a group
Normal subgroup
Ring/commutative ring/ring with unity
Integral domain
Subring
Ideal
Quotient ring
Maximal ideal
Prime ideal
Field
Invertible element of a ring
Zero divisor
Ring homomorphism

Theorems to state(be able to prove *d ones)
*Cayley's Theorem
*ord(a)=|<a>| (Thm 3,4 ch.10)
*If ord(a)=n, then a^k=e iff n/k
Isomorphism of Cyclic Groups
*Every partition determines an equivalence relation and vice versa
*Cosets of a subgroup partition a group
*Lagrange's Theorem
*Kernels of homomorphisms and normal subgroups are the same
*Coset multiplication is well defined on the set of cosets of a normal subgroup and this forms a group
*Fundamental homomorphism theorem
Second isomorphism theorem
Sylow's Theorem
Basis theorem for finite abelian groups
*A ring has the cancellation property iff it has no zero divisors
*The kernel of a ring homomorphism is an ideal
*The image of a ring homomorphism is a subring
*The quotient ring construction makes sense
*The fundamental homomorphism theorem for rings
*If J is a prime ideal of R then R/J is an integral domain
If J is a maximal ideal of R then R/J is a field