Download Definitions Abstract Algebra Well Ordering Principle. Every non

Denitions Abstract Algebra
Well Ordering Principle. Every non-empy set of Z+ contains a smallest element.
Division Algorithm. If a, b ∈ Z and b > 0, (∃!q, r ∈ Z) 3 a = bq + r
with 0 ≤ r < b.
GCD is a linear combination.
(∀a, b ∈ Z \ {0}) 3 b > 0, (∃s, t ∈ Z) 3−→ gcd(a, b) = as + bt
and furthermore gcd(a, b) is the smallest integer of the form as + bt.
Euclid's Lemma. If p is prime and p|ab then p|a or p|b.
Fundamental Theorem of Arithmetic.
(∀z ∈ Z+ ) 3 z > 1, z is prime or a unique product of primes.
First PMI.
Let a ∈ N. Let S ⊆ N with two properties:
i) a ∈ S,
ii) (∀n ∈ N)such that n ≥ a, if n ∈ S −→ n + 1 ∈ S.
Then (∀n ∈ N) 3 n ≥ a, n ∈ S .
Second PMI.
Let a ∈ N. Let S ⊆ N with two properties:
i) a ∈ S,
ii) (∀n ∈ N) such that a ≤ n, ∀k 3 a ≤ k ≤ n, if k ∈ S −→ n + 1 ∈ S.
Then (∀n ∈ N) 3 n ≥ a, n ∈ S .
Equivalence Relations.
Let a, b ∈ S and R ⊆ S × S 3 (a, b) ∈ R:
1) if (∀a ∈ S) (a, a) ∈ R reexive property
2) if (∀a, b ∈ S) (a, b) ∈ R −→ (a, b) ∈ R symmetric property
3) if (∀a, b, c ∈ S) (a, b) ∈ R ∩ (b, c) ∈ R −→ (a, c) ∈ R transitive property
then R is an equivalence relation on S
Equivalence class of a set S containing a: [a] = {x ∈ S | x ∼ a}
Partition of a set S : a collection on non-empty disjoint sets whose union is S
Equivalence classes partition set S :
The equivalence classes of an equivalence relation on a set S constitute a partition
of S .
Conversely, for any partition P of S , there is an equivalence relation on S whose
equivalence classes are the elements of P .
Function (mapping) ϕ from set A (domain) to set B (range):
A rule that assigns to each element a ∈ A exactly one element b ∈ B
1
2
Image of A under ϕ : A → B equals {ϕ (a) | a ∈ A}
Composition of functions φϕ
Given ϕ : A → B and φ : B → C , the mapping from A → C dened
by φϕ (a) = φ (ϕ (a))
One-to-one ϕ : A → B
having the property (∀a, b ∈ A) ϕ (a) = ϕ (b) −→ a = b
Onto ϕ : A → B
having the property (∀b ∈ B) (∃a ∈ A) 3 ϕ (a) = b
Properties of functions α : A → B , β : B → C , γ : C → D
1) (γβ) α = γ (βα) associativity
1−1
1−1
2) if α : A −→
B , β : B −→ C , then βα is 1-1 (injective)
onto
onto
3) if α : A −→
B , β : B −→ C , then βα is onto (surjective)
1−1
1−1
4) if α : A −→ B , then there is a function α−1 : B −→
A, such that
onto onto
−1
−1
(∀a ∈ A) α α (a) = a and (∀b ∈ B) αα
(b) = b
Binary operation (on set G): a function that assigns to each ordered pair of
elements G an element of G, i.e., f : (G × G) → G
closure: the condition that members of an ordered pair from set G combine to
yield a member of G
group: a set G together with a binary operation with the following properties
1) associativity: (∀a, b, c ∈ G) (ab) c = a (bc)
2) uniqueness of the identity: (∀a ∈ G) (∃!e ∈ G) 3 ae = ea = a
3) uniquesness of inverses: (∀a ∈ G) (∃!b ∈ G) 3 ab = ba = e
Abelian: a group having the property that (∀a, b ∈ G) ab = ba
order of a group | G |, the number of elements in G
order of an element | g |, the smallest n ∈ Z+ 3 g n = e
subgroup H of G, H ≤ G: H⊆ G such that H itself forms a group
proper subgroup H of G, denoted H < G
a subgroup of H of group G that is not equal to G
trivial subgroup: {e}
One-step Subgroup Test
If G is a group and H is a nonempty subset of G, then H ≤ G if
(a, b ∈ H) −→ ab−1 ∈ H
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Two-step Subgroup Test
If G is a group and H is a nonempty subset of G, then H ≤ G if
(a, b ∈ H) −→ ab ∈ H ∩ a−1 ∈ H
Finite Subgroup Test
If H is a nite nonempty subset of groupG, then H ≤ G if
H is closed under the operation of G
cyclic (∃a ∈ G) 3 G = hai = {an | n ∈ Z}
hai is the cyclic subgroup of G generated by a
for any element a of group G, hai = {an | n ∈ Z} ≤ G
if hai = G, then a is the generator of G
center of a group, denoted Z (G)
Z (G) = {a ∈ G | (∀g ∈ G) ag = ga} and Z (G) ≤ G
centralizer of a in G, denoted C (a)
C (a) = {g ∈ G | ag = ga} and C (a) ≤ G
(a ∈ G) ai = aj ←→(i) | G |= n < ∞ and n | i − j
if (a ∈ G) 3 ai = aj then
1)| a |=| G |=| hai | and
2) ak = e implies | a | divides k
if (a ∈ G) 3| a |= n, then ai = aj ←→(i) gcd (n, i) = gcd (n, j)
if (a ∈ G) 3 G = hai ∩ | G |= n, then G = ak ←→(i)gcd (n, k) = 1
(k ∈ Zn ) 3 Zn = hki←→(i)gcd (n, k) = 1
Fundamental Theorem of Cyclic Groups
1) every subgroup of a cyclic group is cyclic
2) if | hai |= n then the order of any subgroup of hai is a divisor of n n 3) (∀k ∈ Z+ ) 3 k | n, hai has exactly one subgroup of order k, i.e., a k
For subgroups of Zn only, for each positive divisor k of n,
1) the set hn/ki is a unique subgroup of order k,
2) these are the only subgroups of Zn
For subgroups of Zn only, for a = k, | k |=
n
gcd(n,k)
Green not covered in lectures.
number of elements of each order in a cyclic group
if d is a positive divisor of n group,
the number of elements of order d in a cyclic group of order n is ϕ (d)
number of elements of order d in a nite group
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in a nite group, the number of elements of order d is divisable by ϕ (d)
Euler phi function ϕ (d)
1) ϕ (1) = 1
2) (∀n > 1) ϕ (n) =| U (n) | (the number of positive integers less
than n, and relatively prime to n
subgroup lattice
a partial ordering of all the subgroups of a group
permutation of a set A
a function f : A → A that is a bijection
permutation group of a set A
a set of permutations of A that forms a group under function composition
symmetric group of order n, denoted Sn
the set of all permutations of a set A, | Sn |= n!
products of disjoint cycles
every permutation can be written as a cycle or as a product of disjoint cycles
disjoint cycles commute
if two cycles α and β have no entries in common, then αβ = βα
order of a cycle σ , is the length of the cycle: e.g. if σ = (153) →| σ |= 3
the order of a permutation of a nite set written in disjoint cycle form is the
if | σ |= x∩ | β |= y , then | σβ |=| βσ |= lcm (x, y)
e.g., | (153) (2468) |= lcm (3, 4) = 12
product of 2-cycles
every permutation in Sn where n > 1 is a product of 2-cycles
even (odd) permutation
a permutation that can be decomposed into a product of an even (odd) number
of 2-cycles
the group of even permutations of Sn
alternating group of degree n, denoted An
order of An
for n > 1, | An |=
n!
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