-
What is the one step subgroup test?
Let G be a group and H be a non-empty subset of G. Let  , if then H is a subgroup.
-
What is the two-step 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, i-j 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 orbit-stabilizer 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 p-subgroup 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 p-subgroup 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 p-subgroup of G
-
-
What is the second Isomorphism Theorem for Rings.
If A and B are ideals of R then .
|
|
