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What is the one step subgroup test?
Let G be a group and H be a nonempty subset of G. Let , if then H is a subgroup.

What is the twostep subgroup test?
Let G be a group and H a subgroup of G. Let . If H is closed under inverses and closed under the group operation H is a subgroup.

Define: The center of the group G.
The center of a group G, denoted .

Define: Centralizer of a in G.
The centralizer of , .

Let G be a group, with such that . What can be said about i and j.
If G is infinite i=j. If G is finite, ij divides n.

What is the order of and .
and .

Define: Coset of H in G.
Let be a subgroup of . Let , the coset of H in G is defined as .

State Lagranges Theorem.
If H is a subgroup of the finite group G. Then divides .

Define: .
is the order of .

How many groups of order are they. Where is prime.
1 up to isomorphism, .

What is Fermat's Little Theorem?
Let be prime. Then for all .

How many groups of order 2p are there? Where p is a prime greater than 2
two groups, or .

Define: Let G be a group and be a set. Let . Define the stabilizer of in G.
The stabilizer of in G denoted

Define: Let be a set and be a group acting on . Define the orbit of the point .
The orbit of in , denoted .

What is the orbitstabilizer theorem.
for any , .

Let G and H be finite cyclic groups. When is cylic
If and only if order of G and H are relatively prime.

Define: .

Define: A characteristic subgroup of G.
N is a characteristic subgroup of G if .

Define: Normal subgroup of G.
Let be a subgroup of the group . Then N is normal if and only if . Denoted .

What is the normal subgroup test?
If and then N is normal in G.

Let G be a group and be the center of G. Assume is cyclic, what can be said about G?
G is Abelian.

How many groups of order are there?
two, or .

State the first isomorphism Theorem.
Let be an onto homomorphism of groups, rings or modules. Then .

What is the second isomorphism Theorem for groups.
If K is a subgroup of G and H is a normal subgroup of G, then .

State the Third Isomorphism Theorem.
If M and N are normal subgroups of G and N is a subgroup of M, .

Define: Integral Domain.
An integral domain is a commutative ring with unity and no zero divisors.

Define: Characteristic of a Ring
The least positive integer such that for all . If no such integer exists, the characteristic is said to be zero.

What is the characteristic of an integral domain
either zero or prime.

Define: Ideal of a ring.
A subset of R is an ideal if it is a subring and has the property that for all and .

Test that I is an ideal of R
 1) Check that I is nonempty.
 2) closed under addition.
 3) absorbs elements from R.

Define: Prime and maximal ideals.
An ideal is prime if is multiplicative closed. And ideal M of R is maximal if the only ideal containing M is R.

Let R be commutative. When is and integral domain?
If and only if A is prime.

Let R be commutative. When is a field?
If and only if A maximal.

What is the chinese remainder theorem?
Let R be a ring and I and J be coprime ideals of R, then .

What is the mod p test for irreducibility.
Let f(x) be a polynomial over the integers of degree greater than one. If there exists p, prime such that is irreducible and doesn't change degree then f(x) is irreducible.

What is Eisenstein's Criterion
Let , if there exists a prime p such that , and , then f(x) is irreducible.

When does prime imply irreducible
In an integral domain.

In a UFD is irreducible prime.
Yes.

Let f(x) be a polynomial over the field F. When does f(x) have a multiple zero in an extension E.
If and only if f(x) and f'(x) have a common factor in F[x].

Let f(x) be an irreducible polynomial over K. How many multiple zeros does f(x) have?
If K has characteristic 0, then f(x) has no multiple zeros. If K has characteristic p then f has a multiple zero if it is of the form for some g(x) in K[x].

Define: Perfect Field.
A field K is called perfect if it has characteristic zero or the map defined by is onto.

Let f(x) be an irreducible polynomial over F and E be the splitting field of f(x). What can be said about the multiplicity of the zeros in E.
Every zero has the same multiplicity.

Define the minimal polynomial for a over F.
The minimal polynomial for a over F is the monic polynomial of least degree that has a has a root.

State the tower law.
Let F be a field and E be a finite extension of F. Let K be a finite extension of E. Then K is a finite extension of F and .

State the primitive element theorem.
If F has characteristic zero and with a and b algebraic over F then there exists c such that F(a,b)=F(c)

Define: Conjugacy class of a.
.

Let G be a finite group. How many conjugates does a have in G?

What is the class equation?
For any finite group G, .

What is Sylows First Theorem?
Let G be a finite group and let p be a prime. If divides the order of G then G has at least one subgroup of order .

Define: Sylow psubgroup of the finite group G.
Let p be prime and let k be the largest power of p that divides the order of G. Then any subgroup of order is a sylow psubgroup of G.

What does it mean for two subgroups to be conjugate?
H and K are conjugate if there exists x in G such that .

What is Sylow's second theorem
If H is a subgroup of the finite group G, and the order of H is a power of a prime divisor of G. Then H is contained in some sylow psubgroup of G


What is the second Isomorphism Theorem for Rings.
If A and B are ideals of R then .

