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Gravity
Terms in this set (157)
Cayley's Theorem
Every group G is isomorphic to a (sub)group of permutations on a set.
Normal Subgroup Test
A subgroup H of G is normal in G if and only if xHx⁻¹⊆H for all x in G.
Cauchy's Theorem for Finite Abelian Groups
Let G be a finite Abelian group whose order is divisible by a prime p. Then G contains an element of order p.
|A|
The cardinality of A, the number of elements in A.
One to One
f is one to one if and only if for every a₁, a₂, ∈ A, a₁≠a₂ implies f(a₁)≠f(a₂).
Equivalently: f(a₁)=f(a₂) implies a₁=a₂.
Onto
f is onto if and only if for every element in the codomain there exists at least one element connected.
Equivalently: for all b ∈ B there exists an a ∈ A such that f(a)=b.
Well-Ordering Principle
It says any non-empty subset A ⊆ Z⁺ has a least element, an axiom.
Division Algorithm
Let a, b ∈ Z, b>0, then there exists unique integers q, r such that a= q*b+r and 0≤r≤b.
Theorem 0.2
Let a, b ∈ Z, a≠0, b≠0. Let d=gcd(a, b). Then there are integers (not unique) p, q such that p*a+q*b=d.
Relation
A relation between a set A and a set B is a subset R⊆AxB. R={(a, b)| a∈A is related to b∈B}.
Equivalence Relation
An equivalence relation on a set A is a relation R on A such that:
R is symmetric- aRb implies bRa.
R is reflextive- for all a ∈ A, aRa.
R is transitive- for all a, b, c in A if aRb and bRc, then aRc.
Binary Operation
Under the operation denoted by ab, the set G together with a binary operation is a group if and only if the following properties are satisfied:
Associative: for all a, b, c ∈ G whe have (ab)c=a(bc).
Identity: ∃e ∈ G such that for all a ∈ G ea=ae=a.
Inverse: for all a ∈ G, ∃b ∈ G such that ab=ba=0.
Abelian
A group is Abelian provided that the operation is commutative that is for all a, b ∈ G, ab=ba. To prove something Abelian we must show (ab)ⁿ=aⁿbⁿ
GL(n, R)=General Linear Group
The set of nxn invertible matrices under matrix multiplication.
Cancellation
In any group, the left and right cancellation laws hold. That is if ab=ac, then b=c, and if ba=ca, then b=c.
Note: If ab=ca, id does not follow in general that b=c.
Order of a Group
The number of elements of a group (finite or infinite) is called its order.
Order of an element
The order of an element g in a group G is the smallest positive integer n such that gⁿ=e (in additive notation, this would be ng=0).
Infinite Order
If no such integer exists such that gⁿ=e, then g has infinite order.
Socks-Shoes Identity
Let G be a group a, b∈G. (ab)⁻¹=b⁻¹a⁻¹
U(9)
Take the numbers (0-8) that are relatively prime to 9, {1, 2, 4, 5, 7, 8}.
U(n)
U(n)={a| 1≤a<n and gcd(a, n)=___}
Subgroup
A subgroup H of a group G is a non-empty subset. H⊆G such that when the group operation is restricted to H, H becomes a group.
Subspace Test
If H is a non-empty subset of a group G, what conditions will guarantee that H is a subgroup of G.
Two-Step Test
Let H be a non-empty subset of a group G. If ab ∈ H whenever a, b∈H and a⁻¹∈ H for all a∈H, then H is a subgroup of G.
One-step Test
Let H be a non-empty subset of a group G. If ab⁻¹∈H for all a, b∈H, then H is a subgroup of G.
Finite Subgroup Test
Let G be a group and let H be a finite subset of G. If H is closed under the group operation (multiplication), then H is a subgroup.
Center of G
Denoted Z(g) is the set of all elements in G that commute with all elements of G. Z(G)={a∈G| ab=ba for all b∈G}
The center is abelian, a subset and a subgroup.
If G is Abelian
then Z(G)=G. an element in the center commutes with everything.
Centralizer
Let G be a group, a∈G the centralizer of a in G, denoted c(a) is the set of all elements in G that commute with a.
In symbols, c(a)={b∈G| ab=ba}.
Centralizer is a subgroup.
Cyclic Group
<a>={aⁿ| n∈Z} ⊆ G is the cyclic subgroup of G generated by a.
We say G is cyclic if G=<a> for some a∈G
Let |a|=n, a∈G, and let gcd(k,n).
Then |a^k|=n/d, and <a^k>=<a^d>.
What other than a itself are the generators of <a> if |a|=n.
a^i a generator if and only if |a^i|=n, but |a^i|=n/gcd(n, i)=n
Fundamental Theorem of Cyclic Groups
Every subgroup H of a cyclic group <a> is cyclic. Moreover, if |<a>|=n, then the order of any subgroup of <a> is a divisor of n, and for such positive divisors d|n, there is a unique subgroup of <a> of order d, namely <a^(n/d)>.
<a^k>
=<a^gcd(n, k)>.
Theorem 4.3
Every subgroup of G is cyclic, having the form H=<a^m>. If |a|=n=|G| if finite <∞, then |a^m|=n/gcd(m, n)=d, which divides n.
Moreover, for every d|n, we have a unique subgroup of order d: <a^(n/d)>.
If |a|=n, so <a>=g has order n
then the number of elements of G of order d, where d|n, is phi(d)
In a finite group G, the number of elements of order d
is a multiple of phi(d)
You can find other generators by raising the original generator
to a number that is relatively prime to the generator.
Permutation
A permutation of a set A is a 1-1 and onto function G:A→A.
In general permutations do not commute. σα≠ασ.
A Permutation Group
Is a set of permutations on A that form a group under function computation.
Symmetric Group
The symmetric group on n letters, denoted Sn is the group of all permuations σ:{1,...,n}→{1,...,n}
|Sn|
=n!
Properties of Disjoint Cycles
The product of disjoint cycles is commutative.
Disjoint cycles commute.
The order of a permutation
=lcm (lengths of disjoint cycles that represent it.)
Theorem 5.4
Every permutation of {1, 2,..., n} can be expressed as a product of (not necessarily disjoint) 2-cycles.
Suppose σ= c₁c₂...cs and σ=d₁d₂...dt, where the ci and dj are 2 cycles.
Then s and t are either both even or both odd. [On s-t even.]
If we compose a permutation with a 2-cycle the number of inversions
changes by an odd number.
An inversion occurs when
ai>aj for i<j. Last time we saw that when one composes G with a 2-cycle the total number of inversions changes by an odd number.
If the identity is equal to a product of 2-cycles
then the number of 2-cycles is even.
If a permutation, say G, is expressed as a product of 2-cycles in 2 ways, say σ=c₁c₂...ck, σ=d₁d₂...dl
then k≡l (mod 2)
σ if even (odd) if and only if
σ is represented by an even (odd) number of 2-cycles.
Some authors define the sign parity of σ to be 1. σ is even, -1 if σ is odd.
1 if σ is even
-1 if σ is odd
Alternative group.
An={σ ∈ Sn | σ is even|. This is called the alternating group.
An≤Sn.
If n is odd
(n__) is even and vice versa.
If H ≤ Sn
then either H contains only even permutations, or exactly half are even and half are odd.
Two groups are isomorphic
if they are "essentially the same." That is, the groups can be relabeled so that their cayley tables are identicle.
Isomorphism
An isomorphism from a group G to a group G' is a one to one and onto map. phi: G→G' that "preserves the multiplication": that is, for all a, b∈G, we have that phi(ab)=phi(a)*phi(b).
If phi:G→G' is an isomorphic of groups
then phi⁻¹: G⁻¹→G is an isomorphism of groups.
phi⁻¹ is 1-1 and onto.
Automorphism
An automorphism of a group is an isomorphism phi:G→G.
Aut(G)
is a group under composition of isomorphisms.
phi₁, phi₂ ∈ Aut(G) G-(phi₁)->G-(phi₂)->G
Aut(g) is a group under composition.
α preserves the group operation if and only if
G is abelian.
Inner Automorphisms induced by a
Let G be a group, and let a∈G. The function phia defined by phia(x)=axa⁻¹ for all x in G is called the inner automorphism of G induced by a.
Aut(G) and Inn(G) are groups
The set of automorphisms of a group and the set of inner automorphisms of a group are both groups under operation of function composition.
Properties of Inn(G)
Inn(G) is the set of all phia={phia | a∈G}
Inn(G)⊆Aut(G)
Theorem 6.5
Aut(Zn)≈U(n)
Let S be a set: A relation R on S (R⊆SxS) is an equivalence relation if and only if
R is reflexive, symmetric, and transitive.
If R is an equivalence relation on S and a∈S then the equivalence class of a is [a]=
{b∈S | aRb}.
The equivalence class forms a partition of S
That is [a]≠∅ for any a, S=U[a] and if [a]≠[b], [a]∩[b]=∅, a∈S.
Right Coset
{ha | h∈H}=Ha.
Properties of Cosets
Let H be a subgroup of G, and let a and b belong to G. Then,
1. a∈ah
2. aH=H if and only if a∈H
3. (ab)H=a(bH) and H(ab)=(Ha)b.
4. aH=bH if and only if a∈bH
5. aH=bH or aH∩bH=∅
6. aH=bH if and only if a⁻¹b∈H
7. |aH|=|bH|
8. aH=Ha if and only if H=aha⁻¹
9. aH is a subgroup of G if and only if a∈H.
Key Properties of Cosets
Either aH=bH or aH∩bH=∅ (same for right cosets).
|aH|=|H| for all a∈G
Lagrange's Theorem
Let G be a finite group, and H≤G then |H| | |G|
Index
The index of H in G is |G|/|H|= the number of distinct cosets (left or right). The notation for the index is |G : H|.
Corallary 1 for Lagrange's Theorem:
|G : H|=|G|/|H| if |G|<∞.
Corallary 2 for Lagrange's Theorem:
Let a∈G where G is finite. |a| | |G|. |a|=|<a>|, and |<a>| | |G|.
Corallary 3 for Lagrange's Theorem:
Groups of prime order are cyclic.
Corallary 4 for Lagrange's Theorem:
|G|<∞. For all a∈G, a^|G|=e.
Fermat's Little Theorem:
For every integer a and every prime p we have a^p≡a (mod p).
Converse of Lagrange's Theorem:
If m | |G|, then there is a subgroup H≤G of order m. This is not true in general.
For finite cyclic groups G=<a>, if m | |<a>|,
then there is a unique subgroup of order m.
Let G be a group, and H, K subgroups of G. We define HK={hk | h∈H, k∈K}
Where H*K is not necessarily a subgroup of G. (It is if G is Abelian).
If H, K are finite subgroups of G,
then |H*K|=(|H|*|K|)/|H∩K|.
Let G be a group with order |G|=2p, p is prime,
then G is isomorphic to either Z2 or Dp.
Classification of groups of order 2p, p is prime, p>2
Let G be a group, |G|=2p, p is prime p≥3, then either G≈Z2p or G≈Dp.
Stabilizer
Let x∈X, then the stabilizer of x in G is the subset {α∈G | α(x)=α} denoted StabG(x).
StabG(x)≤G
Orbit
The orbit of x is the set of all g∈x such that g=α(x) for some α∈G. The orbit is denoted as orbG(x) and can be defined as orbG(x)={α(x) | α∈ G}.
Let G be a finite group of permutations of a set x,
then for every x∈X, we have that |G|=|orbG(x)|*|StabG(x)|.
External Direct Product
Let G₁, G₂ be groups. Then the external direct product of G₁, G₂ is G₁⊕G₂={(a, b) | a∈G₁, b∈G₂}
Cartesian product + binary operation:
(a₁, b₁)*(a₂, b₂)=(a₁a₂, b₁b₂)
(G₁⊕G₂)⊕G₃≈G₁⊕(G₂⊕G₃)
((a₁, b₁), c₁) if and only if (a₁, (b₁, c₁)) if and only if (a₁, b₁, c₁).
G₁⊕G₂⊕G₃
G₁⊕...⊕Gn
If (g₁,...,gn) ∈ G₁⊕G₂⊕...⊕Gn and the Gi are all finite,
then |(g₁, g₂,...,gn)|=lcm(|g₁|, |g₂|,..., |gn|).
Let G, H be finite groups, then G⊕H is cyclic if and only if
G, H are both cyclic and gcd(|G|, |H|)=1
If something is cyclic then
the order of the group equals the order of the elements.
G⊕H is Abelian if and only if
G, H both are Abelian.
If s, t are relatively prime integers
then U(st)≈U(s)⊕U(t)
Closed Form
S(n)=(3ⁿ-1)/2
Recurrance
S(n)={1 n=1, 3*S(n-1)+1 for n>1
Normal
Let G be a group H<G. We say H is normal in G, and write H∠G if and only if for all a∈G, aH=Ha.
A subgroup H of G is normal if and only if
for all a∈G, aHa⁻¹⊆H.
An⊆Sn is normal.
...
Why do we care about normal subgroups?
The set of cosets of a normal subgroup form a group.
Factor Group
When the subgroup H of G is normal, then the set of left/right cosets of H in G is itself a group, the factor group of G by H.
(aH)⁻¹=a⁻¹H. We denote this group G/H.
Let G be a group Z(G) be a coset of G₁ (We know Z(g) is normal to G). If G/Z(G) is cyclic,
then G is Abelian. [Then Z(G)=G, therefore G/Z(G)={e}]
Let G be a group, Z(g) its center. (We have seen that Z(G) is normal to G),
then G/Z(G)≈Inn(G). Let H=Z(G).
Coset
A coset of a subgroup H≤G, the set aH is called the left coset of H∈G containing a, while Ha is called the right coset of H∈G containing a.
External Direct Product
The external direct product of G₁, G₂,..., Gn, is written as G₁⊕G₂⊕...⊕Gn is the set of all n-tuples for which the ith component is an element of Gi and the operation is component-wise. In symbols G₁⊕G₂⊕...⊕Gn={(g₁, g₂,..., gn)|gi∈Gi}
Stabilizer of a Point
Let G be a group of permutations of a set S. For each i in S, let stabG(i)={phi∈G | phi(i)=i}. We call stabG(i) the stabilizer of i in G.
Orbit of a Point
Let G be a group of permutations of a set S. For each i in S, let orbG(i)= { phi(i) | phi∈G}. The set orbG(i) is a subset of S called the orbit of i under G. We use |orbG(i)| to denote the number of elements in orbG(i).
U(st)≈U(s)⊕U(t) if gcd(s, t)=1,
then Us(st)≈U(t).
Proposition of normal
H is normal in G if and only if for all a ∈G, for all h∈H, ∃h'∈H such that ah=h'a
Theorem 9.1 A Subgroup H of G is normal if and only if
for all a∈G, aHa⁻¹⊆H
Associativity
((aH)(bH))(cH)=(ah)(bHcH), (abH)(cH)=(ah)(bcH), ((ab)c)H=a(bc)H
Inverses
(ah)⁻¹=a⁻¹H, we denote this group G/H. We call it the factor group (or quotient group) of G by H.
Theorem 9.2: Let G be a group, Z(g)=coset of G (we know Z(G)∠G. If G/Z(G) is cyclic then
G is Abelian [then Z(G)=G, Therefore G/Z(G)={e}
Theorem 9.4: Let G be a group, Z(G) its center. (We have seen that Z(G)∠G)
then G/Z(G)≈Inn(G). Let H=Z(G).
Theorem 9.5: (Cauchy's Theorem for Abelian Groups):
Let G be a finite Abelian group, and p a prime such that p| |G| then G has an element of order p.
Internal Direct Products
Let G be a group and H₁H₂ normal subgroups of G. We know already that H₁H₂={h₁h₂ | h₁∈H₁, h₂∈H₂} is a subgroup. (We only need one of H₁, H₂ to be normal.) Suppose H₁H₂=G and H₁∩H₂={e}, then G≈H₁⊕H₂
If H₁, H₂∠G, H₁H₂=G and H₁∩H₂={e},
then phi:H₁⊕H₂→G, (h₁, h₂)→h₁h₂, is an isomorphism in this case G=H₁xH₂ id the internal direct product of H₁ and H₂.
Theorem 9.6 If we have a series of normal subgroups H₁, H₂, ..., Hn∠G such that H₁∩H₂={e} H₁H₂∩H₃={e}... and G=H₁H₂...Hn,
then phi:H₁⊕H₂⊕...⊕Hn→G is an isomorphism (h₁, h₂,..., hn)→h₁h₂,...,hn we write G=H₁xH₂x...xHn.
Theorem 9.7: Let G be a group, |G|=p², p is prime,
then either G≈Z_p² or G≈Z_p⊕Z_p.
Homomorphism
from a group G to a group H is a map phi: G→H such that phi(ab)=phi(a)phi(b) for all a, b∈G.
Theorem 10.1: Properties of Elements Under Homomorphisms:
Let phi be a homomorphism from a group G to a group G' and let g be an element of G, then:
1. phi carries the identity of G to the identity of G'.
2. phi(gⁿ)=(phi(g))ⁿ for all n in Z.
3. If |g| is finite, then |phi(g)| divides |g|.
4. Ker(phi) is a subgroup of G.
5. phi(a)=phi(b) if and only if aker(phi)=bker(phi).
6. If phi(g)=g', then phi⁻1(g')={x∈G | phi(x)=g'}=gker(phi).
Theorem 10.2: Properties of Subgroups Under Homomorphisms:
Let phi be a homomorphism from a group G to a group G' and let H be a subgroup of G, then:
1. phi(H)={phi(h) | h∈H} is a subgroup of G'.
2. If H is cyclic, then phi(H) is cyclic.
3. If H is Abelian, then phi(H) is Abelian.
4. If H is a normal in G, then phi(H) is normal in phi(G).
5. If |ker(phi)|=n, then phi is an n to 1 mapping from G onto phi(G).
6. If |H|=G, then |phi(H)| divides n.
7. If k- is a subgroup of G-, then phi⁻¹(k-)={k∈G | phi(k)∈k-} is a subgroup of G.
8. If k- is a normal subgroup of G-, then phi⁻¹(k-)={k∈G | phi(k)∈k-} is a normal subgroup of G.
9. If phi is onto and ker(phi)={e}, then phi is an isomorphism from G to G-.
A homomorphism of a group
Is a map phi: G→H such that phi(ab)=phi(a)phi(b) for all a, b∈G. Ker(phi)={a∈G | phi(a)=e}=phi⁻¹({e}).
Proposition of homomorphisms: If phi: G→H is a homomorphism and H'∠H,
then phi⁻¹(H')∠G.
We must first prove phi⁻¹(H')∠G, then prove H' is normal in H.
Corollary of homomorphisms:
Let phi: G→H be a homomorphism, then ker(phi)∠G.
Theorem (First Isomorphism Theorem):
Let phi:G→H be a homomorphism, then the map phi-:G/ker(phi)→phi(G)≤H. Is an isomorphism, aker(phi)→phi(a).
Corollary: If phi is a homomorphism, phi:G→H is onto (H=phi(g)),
then G/ker(phi)≈H.
Theorem 11.1: Every finite Abelian group is a direct product of cyclic groups of prime power order.
(G≈Z_p₁ⁿ¹⊕...⊕Z_p₂ⁿ²) the p_i are not necessarily distinct. Moreover, the primes p_i in the decomposition and the powers n_i are uniquely determined by the group.
One can systematically list all the isomorphic types of Abelian groups of given order.
...
Lemma 1 of Theorem 11.1: Let G be a finite Abelian group of order pⁿn, where gcd(p, n)=1,
then G=HxK, where H={x∈G | x^pⁿ=e} and k={x∈G | xⁿ=e}.
More over |H|=pⁿ.
Corollary 1 of Theorem 11.1: G=H₁xH₂x...xH_m, |H_i|=p₁ⁿ¹p₂ⁿ...(p_m)^nm
then H_i are not necessarily cyclic.
Corollary 2 of Theorem 11.1: Existence of subgroups of Abelian Groups if m divides the order of a finite Abelian group G,
then G has a subgroup of order m.
Lemma 2 of Theorem 11.1: Let |G|=pⁿm G is finite Abelian. Let a∈G be an element of maximum order, |a|=p^r, r≤n,
then G=<a>xk where k∠G, k is a subgroup that we will find, it's not the same k as the previous lemma, call it L.
Lemma 3 of Theorem 11.1: G is a finite Abelian group
then G=H₁xH₂x...xH_k (lemma 1), (Z_p₁)^s₁⊕(Z_p₁)^s₂⊕...⊕(Z_p₁)^s_l.
Therefore G≈ direct product of cyclic groups of prime power order and internal.
Lemma 4 of Theorem 11.1: G is a finite Abelian group of prime power order p^r. If G=H₁xH₂x...xH_m=k₁xk₂x...xk_n where |H_i|≥|H_i+1| and |k_j|≥|k_j+1|,
then m=n and |H_i|=|k_j|.
Definition of a Ring
A ring is a set R with two operations +, *, addition and multiplication, such that R is an Abelian group under addition and multiplication distributes over addition "on both sides."
Axion for Ring R:
1. a+b=b+a for all a, b∈R.
2. (a+b)+c=a+(b+c) for all a, b∈R.
3. There exists O∈R such that O+a=a+O for all a∈R.
4. For all all ∈R, ∃(-a) such that a+(-a)=0.
5. (ab)c=a(bc).
6. a(b+c)=ab+ac, (b+c)a=ba+ca.
Another definition of a ring R.
A ring is a set with two associative binary operations called addition (a+b) and multiplication (a*b=ab) such that R is an Abelian group under addition and multiplication distributes over addition on both sides.
Note:
There may or may not be a multiplicative identity, and there may be non-zero elements that have no multiplicative inverse. In addition, multiplication need not be commutative. If it IS commutative R is called a commutative ring. Clearly R under addition, being an Abelian group, has all the previously established properties of an Abelian group, there is no need to reprove them in the case of rings.
Subring
is a subset of a ring that becomes a ring itself under the operation of the previous ring.
Definition of subring:
If R is a ring, then a subset of S⊆R is a subring if the addition and multiplication operations when restricted to s make s into a ring.
One step test for "subringhood": A non-empty subset s of runt R is a subring if and only if
for all a, b∈S, we have a-b∈S and a, b∈S.
Theorem 12.1: Let a, b∈R, then:
1. a0=0a=0
2. a(-b)=(-a)b=-(ab), additive inverse of b.
3. (-a)(-b)=ab
4. a(b-c), which means b+(-c)=ab-ac.
5. If R has a unity 1(-1)a=-a.
6. If R has a unity 1(-1)(-1)=1.
Theorem 12.2: Suppose 1 and 1' are multiplicative identities (unities) in a ring,
then 1=1. Also the multiplicative inverse a⁻¹ of a, if it exists is unique.
Definition of unity in R:
If a∈R, a≠0, and ab=1 for some h∈R, then we call a unit in R.
Definition of zero-divisor:
Let R be a ring. If a∈R, a≠0, and ab=0 for some b≠0, we call a (and b) a zero-divisor.
Definition of an integral domain:
An integral domain is a commutative rung with unity that has no zero-divisors.
Theorem 13.1 Cancellation: Let R be an integral domain. If a∈R, a≠0,
then ab=ac=b=c. You can "cancel" a.
Definition of a field:
A field is a commutative ring with unity in which every non-zero element is a unit. (Every non-zero element has a multiplicative inverse.)
Proposition of a field:
Every field is an integral domain.
Theorem 13.2: Every finite integral domain
is a field.
Another definition of integral domain:
An integral domain is a commutative ring with unity, and no zero divisors equivalent to a≠0, ab=ac implies b=c.
Another definition of field:
A field R is a commutative rung with unity such that every a≠0 in R has a multiplicative inverse: a(a⁻¹)=1.
Note: Every field is an integral domain, but not vice versa.
...
Definition of characteristic of a ring R:
The characteristic of a ring R is the least positive integer n such that x+x+x+...+x=n*x=0 for all x∈R, or 0. If such an n doesn't exist.
Ex. Z has characteristic 0.
Z₆≈Z₂⊕Z₃ which has characteristic 6.
Theorem 13.3: If R is a ring with unity
then the characteristic of R is 0 if no multiple of 1 is 0 or else it is the least positive n such that 1+1+...+1=n*1=0.
Theorem 13.4: If R is an integral domain (including all fields)
then the characteristic of R is either 0 on a prime p. (Char. R.)
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