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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.
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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.
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Define: The center of the group G.
The center of a group G, denoted .
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Define: Centralizer of a in G.
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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.
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What is the order of and .
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Define: Coset of H in G.
Let be a subgroup of . Let , the coset of H in G is defined as .
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State Lagranges Theorem.
If H is a subgroup of the finite group G. Then divides .
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How many groups of order are they. Where is prime.
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What is Fermat's Little Theorem?
Let be prime. Then for all .
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How many groups of order 2p are there? Where p is a prime greater than 2
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Define: Let G be a group and be a set. Let . Define the stabilizer of in G.
The stabilizer of in G denoted
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Define: Let be a set and be a group acting on . Define the orbit of the point .
The orbit of in , denoted .
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What is the orbit-stabilizer theorem.
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Let G and H be finite cyclic groups. When is cylic
If and only if order of G and H are relatively prime.
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Define: A characteristic subgroup of G.
N is a characteristic subgroup of G if .
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Define: Normal subgroup of G.
Let be a subgroup of the group . Then N is normal if and only if . Denoted .
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What is the normal subgroup test?
If and then N is normal in G.
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Let G be a group and be the center of G. Assume is cyclic, what can be said about G?
G is Abelian.
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How many groups of order are there?
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State the first isomorphism Theorem.
Let be an onto homomorphism of groups, rings or modules. Then .
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What is the second isomorphism Theorem for groups.
If K is a subgroup of G and H is a normal subgroup of G, then .
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State the Third Isomorphism Theorem.
If M and N are normal subgroups of G and N is a subgroup of M, .
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Define: Integral Domain.
An integral domain is a commutative ring with unity and no zero divisors.
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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.
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What is the characteristic of an integral domain
either zero or prime.
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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 .
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Test that I is an ideal of R
- 1) Check that I is nonempty.
- 2) closed under addition.
- 3) absorbs elements from R.
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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.
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Let R be commutative. When is and integral domain?
If and only if A is prime.
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Let R be commutative. When is a field?
If and only if A maximal.
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What is the chinese remainder theorem?
Let R be a ring and I and J be coprime ideals of R, then .
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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.
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What is Eisenstein's Criterion
Let , if there exists a prime p such that , and , then f(x) is irreducible.
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When does prime imply irreducible
In an integral domain.
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In a UFD is irreducible prime.
Yes.
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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].
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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].
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Define: Perfect Field.
A field K is called perfect if it has characteristic zero or the map defined by is onto.
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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.
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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.
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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 .
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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)
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Define: Conjugacy class of a.
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Let G be a finite group. How many conjugates does a have in G?
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What is the class equation?
For any finite group G, .
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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 .
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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.
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What does it mean for two subgroups to be conjugate?
H and K are conjugate if there exists x in G such that .
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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
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What is Sylow's Third Theorem?
- Let p be a prime and let where p does not divide m. Then the number of sylow p-subgroups of G satisfies the follow properties
- 1)
- 2)
- Futhermore all sylow p-subgrops are conjugate.
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What is the second Isomorphism Theorem for Rings.
If A and B are ideals of R then .
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