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Terms in this set (45)
Group
A pair (G, *) such that
1) G is a set and * is a binary operation on G
2) * is associative
3) There exists e in G s.t. x
e=e
x=x (for all x in G)
4) For each x in G, there exists y s.t. x
y=y
x=e
Cyclic
A group is called this if there is an element x in G such that G={x^n | n in Z} and x is called a generator of G. G=<x> and x^n=e and n is the order.
Order
The number of elements in a group G, denoted |G|
Subgroup
A subset of a group G that satisfies:
1) h1*h2 is in H for all h1, h2 in H
2) The pair (H, *) is a group
Center
a subset of a group G such that
Z(G)={z in G | xz=zx for all x in G}. (in other words z commutes with all elements in G) Then Z(G) is a subgroup of G.
Surjective/Onto
If for each t in T, there exists s in S such that f(s)=t
One to One/Injective
If for any two distinct s1, s2 in S, f(s1) does not equal f(s2)
Bijective
when a function is both onto and one-to-one
Image
If f: S --> T is a function then the subset {f(s) | s in S} contained in T is the ______ of f
R-cycle
the function that maps x1 to x2, etc and fixes all other elements of [n] in Sn
Equivalence Relation
We call R on S an ____________ if it satisfies:
1) reflexivity: sRs for all s in S
2) symmetry: For all a,b in S, if aRb then bRa
3) transitivity: For all a,b,c in S, if aRb and bRc then aRc
Equivalence Class
Let R be a relation on S. For any s in S, define this to be s= {x in S | xRs} (s bar on top)
Right Coset
If H is a subgroup of G then a ______ of H in G is a subset of G of the form: Ha= {ha | h in H} for any a in G
Left Coset
If H is a subgroup of G then a ______ of H in G is a subset of G of the form: aH= {ah | h in H} for any a in G
Lagrange's Theorem
Let G be a finite group and let H be a subgroup. Then the order of H divides the order of G. |G|/|H|
Index
If G is a group and H is a subgroup of G, then the ______ of H in G is denoted [G:H] and is the number of distinct right cosets of H in G. If G is finite then, [G:H]= |G|/|H|
Fermat's Theorem
Let p be a prime number. Let a be in Z such that (a,p)=1. Then a^(p-1) = 1 (mod p)
Normal Subgroup
A subgroup H in G is ______ if for every h in H and g in G we have ghg^-1 in H
Define gHg^-1= {ghg^-1 | h in H}
equivalently gHg^-1=H for all g in G
and gH=Hg for all g in G
Normal Subgroup 2
any group of index 2 is
Quotient Group
Let H be a normal subgroup of G. Then G/H is defined by the set of right cosets of H in G defined by: If Hg1, Hg2 in G/H then Hg1*Hg2= Hg1g2
|G/H|= [G:H]= |G|/|H|
Group Homomorphism
Let G and H be groups. Then a _________ from G to H is a function phi: G --> H such that for every a, b in G we have phi(ab)=phi(a)phi(b)
Group Isomorphism
a bijective group homomorphism
Group Automorphism
An isomorphism from a group G to itself
phi: G ---> G
Kernel
Let phi: G---> K be a homomorphism. Then the ______ of phi is defined by
{g in G | phi(g)= ek }. This is a subgroup of G.
First/Fundamental Isomorphism Theorem for Groups
Let phi: G ---> K be a homomorphism. Suppose phi is onto (subjective). Then K is isomorphic to G/ker(phi).
Fundamental Theorem of Finite Abelian Groups
Let G be a nontrivial finite abelian group. Then G is isomorphic to a product of cyclic groups, each of prime order.
For example Z30 is iso to Z2 x Z3 x Z5
Ring
A ___ is a set R, together with two binary operations satisfying:
1) (R, +) is an abelian group
2) mult operation satisfies
a) associative
b) commutative
c) it has an identity element r in R s.t. rr'=r' for all r' in R
3) Distributivity: a(b+c)= ab+ac
Zero Divisor
An element in a ring is a _______ if there exists r' in R with r' not equal to 0 such that rr'=0
Nilpotent
r in R is ______ if there exists n>0 such that r^n=0
Unit
r in R is a _____ if there exists r' in R such that rr'=1
Trivial Ring
Any ring consisting of only one element. So R={x} where x=0R=1R
Domain
This is a ring in which the only zero-divisor is 0
Field
A ring is called this if 0R does not equal 1R (identities) and if every non-zero element of R is a unit
Ideal
a subset I in a ring R is an ______ of R if:
a) (I, +) is a subgroup of (R, +) and
b) for every r in R and s in I, rs in I
Trivial Ideal
{0} is the ______. which is also a prime ideal
Improper Ideal
when R is an ideal in R (I=R)
Proper Ideal
when I in R but does not equal all of R
Prime Ideal
An ideal in R is __________ if:
a) I is a proper ideal
b) if a,b in R and ab in I, then either a in I or b in I
Maximal Ideal
An ideal in R is _______ if:
a) I is a proper ideal
b) there is no proper ideal J in R such that I is contained in J and J is contained in R
ex. Z12 has proper ideals: {0}, 2Z12, 3Z12, 4Z12, and 6Z12 but only 3Z12 and 2Z12 are maximal because the other two are contained within the two maximal ones
Ring Homomorphism
a function from R to S (phi: R---> S) where R, S are rings such that:
1) phi(a+b)= phi(a)+phi(b)
2) phi(ab)= phi(a)phi(b)
3) phi(1R)= 1S (one gets sent to one)
for all a,b in R
Ring Isomorphism
A bijective ring homomorphism
Ring Automorphism
A ring isomorphism from a ring to itself
Quotient Homomorphism
Let I in R be an ideal. Then R/I is a ring. There is a ring ___________:
phi: R---> R/I (a --> I + a) such that
phi(ab)= I +ab= (I+a)(I+b)= phi(a)phi(b) and
phi(1R)= I + 1R = 1R/I
Ring Kernel
If phi: R---> S is a ring homomorphism, define the _____ of phi to be
{r in R | phi(r)= 0R}
First Isomorphism Theorem for Rings
If phi: R--> S is an onto homomorphism of rings, then R/ker(phi) to isomorphic to S.
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