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MATH 151 Final Exam
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Gravity
Terms in this set (28)
Division Algorithm
Let a and b be integers with b>0. Then there exists unique integers q and r with the property that a=bq+r, where 0<=r<b
greatest common divisor
A positive integer d such that d is a common divisor of a and b and if d' is any other common divisor of a and b, then d' | d. d = gcd(a,b)
Symmetry Group
the group of all transformations under which the object is invariant with composition as the group operation.
subgroup
A subset H of G such that when the group operation of G is restricted to H, H is a group in its own right. Satisfy these conditions:
1. the identity e of G is in H.
2. if a, b are in H, then ab is in H.
3. if h is in H, its inverse is in H.
cyclic group/subgroup
for a in G, we call <a> the ______ _________ generated by a (where <a>={a^k : k in Z}).
If G contains some element a such that G=<a>, then G is a ________ ______________.
Permutation
a _________ of a set S={A,B,C} is a one to one and onto map π: S -> S.
Even and Odd Permutations
its _____ if it can be expressed as an even number of transpositions
its _____ if it can be expressed as an odd number of transpositions
transpositions
(a,b,c,d,e) = (a,e)(a,d)(a,c)(a,b)
alternating group
the group of even permutations of a finite set
Two Line/One Line/Cycle Notation
1 2 3 4
2 3 4 1
2 3 4 1
(1 2 3 4)
bijection
A map that is both one-to-one and onto
normal subgroup
A subgroup H of a group G is ________ if gH = Hg for all g∈G.
factor group
If N is a normal subgroup of a group G, then the cosets of N in G form a group G/N under the operation (aN)(bN)=abN.
simple groups
A group with no proper nontrivial normal subgroups.
Homomorphism
Between two groups G and G', __________ is a map ϕ: G -> G' such that
ϕ(g1⋅g2)=ϕ(g1)∘ϕ(g2)
kernel
Let ϕ:G→H be a group homomorphism and suppose that e is the identity of H. Then ϕ−1({e}) is a subgroup of G called ____ = {r∈R:ϕ(r)=0}.
Ring
A non empty set R that has two binary operations, addition and multiplication, satisfying:
1. a + b = b + a, for a,b∈R.
2. (a + b) + c = a + (b + c) for a,b,c∈R.
3. There is an element 0 in R such that a + 0 = a for all a∈R.
4. For every element a∈R, there exists an element −a in R such that a+(−a)=0.
5. (ab)c=a(bc) for a,b,c∈R.
6. For a,b,c∈R,
a(b + c) = ab + ac
(a + b)c = ac + bc.
Communitive Ring
A ring R for which ab=ba for all a,b∈R.
Unit
for each a∈R with a≠0, there exists a unique element a^−1 such that (a^−1)a=a(a^−1)=1
Unity
A ring R has ______ if there is an element 1∈R such that 1≠0 and 1a=a1=a for each element a∈R
Subring
A ________ S of a ring R is a subset S of R such that S is also a ring under the inherited operations from R.
ideal
An _______ in a ring R is a subring I of R such that if a is in I and r is in R, then both ar and ra are in I; that is, rI⊂I and Ir⊂I for all r∈R
prime ideal
A proper ideal P in a commutative ring R is called a _______ _______ if whenever ab∈P, then either a∈P or b∈P.
maximum ideal
A proper ideal M of a ring R is a ________ _________ of R if the ideal M is not a proper subset of any ideal of R except R itself.
Integral Domain
A commutative ring with identity is said to be an _______ ________ if it has no zero divisors.
Field
A commutative division ring
Factor/Quotient Ring
Let I be an ideal of R. The factor group R/I is a ring with multiplication defined by
(r + I)(s + I)=rs + I.
Ring Homomorphism
A map ϕ:R→S satisfying
ϕ(a+b)=ϕ(a)+ϕ(b)ϕ(ab)=ϕ(a)ϕ(b)
for all a,b∈R.
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