1. Theories
1.1. lec1
1.1.1. لا توجد اي نظريات في هذه المحاضرة
1.2. lec2
1.2.1. Th1: this proves a subgroup
1.2.1.1. closed
1.2.1.2. inverse
1.2.2. Th2: proves subgroup with 1 conditon
1.2.2.1. inverse
1.2.3. في النظريات اللي تخص ال subgroup بنثبت الاربع شروط والاتجاهين <=>
1.2.4. Th3
1.2.4.1. if G is group and H is subgroup of G then the following is equivalent
1.2.4.1.1. 1- H * X = H * Y : X, Y ∈ G`
1.2.4.1.2. 2- ∃ h ∈ H : y = h * x
1.2.4.1.3. X * Y ^-1 ∈ H
1.3. lec3
1.3.1. Lagrange Theorem
1.3.1.1. Let
1.3.1.1.1. h ∈ H.a
1.3.1.1.2. a ∈ G
1.3.1.1.3. f(h) = h . a
1.3.1.2. we prove that F is bijective
1.3.1.2.1. 1 - 1
1.3.1.2.2. onto
1.3.1.3. | H | = | H . a | = n H. a1, H.a2 , H.at (right district cosets
1.3.1.3.1. |H.a1| ⋃ |H.a2| ⋃ ....... | H.at| = G
1.3.1.3.2. |H.a1| ⋂ |H.a2| ⋂ ....... | H.at| = Ø
1.3.1.3.3. |G| = |H.a1| + |H.a2| + ....... | H.at|
1.3.2. corollaly
1.3.2.1. let a ∈ G => <a> = H then H is cyclic subgroup of G |H| | |G| , |a| = |H|, |a| | |G|
1.3.3. Theorem
1.3.3.1. the intersection of two subgroups is subgroup
1.3.3.1.1. e ∈ H ⋂ K ≠ Ø
1.3.4. Theorem
1.3.4.1. for any subgroup if index = 2 then it's Normal subgroup
1.3.4.1.1. بثبت فيه ان ال right co-sets = left cosets
1.3.4.1.2. notice that في تاني تعويض لما بقول ان ال H.G = H.g ⋂ G
1.3.4.1.3. Proof
1.4. lec4
1.4.1. Theorem
1.4.1.1. for a subgroup these are hold
1.4.1.1.1. H⋂ K is a subgroup
1.4.1.1.2. if H ⊲ G , H ⋂ K ⊲ K
1.4.1.1.3. if H ⊲ G, K ⊲ G , H ⋂ K ⊲ G
1.4.1.2. Proof
1.4.1.2.1. closure
1.4.1.2.2. inverse
1.4.1.2.3. closure & inverse in one step
1.4.2. lemma center is Normal subgroup
1.4.2.1. we prove that it's a center notrmal sub group
1.4.2.2. i.e , x . g = g .,x
1.4.2.3. we let x, y and x = g.x.g^-1 and same for y
1.4.2.4. we prove it's a subgroup by inverse theorem (x.y^-1) =
1.4.3. lemma Normalizer is subgroup of G
1.4.3.1. بثبت انه يحتوي علي ال identity
1.4.3.2. بثبت انه فيه عنصرين x, y بحيث
1.4.3.2.1. x = a * x * a ^ -1
1.4.3.2.2. y = a * y * a ^-1
1.4.3.3. بثبت ان (x.y ^-1 ) تنتمي لل normalizer
1.4.3.4. بثبت ان X . Y ^ -1 * a = a * X , Y ^-1
1.5. lec5
1.5.1. Homomorphism proof
1.5.1.1. F ( a * b ) = F (a) * ` F(b)
1.5.2. Theorem
1.5.2.1. let F: (G, *) b --> (G`, *) be homomorphism then
1.5.2.1.1. F(e) = e`
1.5.2.1.2. f(x^-1) = [f(x) ] ^ -1
1.5.2.1.3. ker f ⊲ G
1.5.3. If f is homomorphic then the following is True
1.5.3.1. f(e) = e`
1.5.3.2. f(x^-1) = [f(x) ] ^-1
1.5.3.3. Ker f ⊲ G
1.5.3.3.1. we prove ker f is subgroup
1.5.3.3.2. we prove Ker f is normal subgroup
1.6. lec6
1.6.1. Theorem
1.6.1.1. let F: (G, *) ==> (G`, *`) be homomorphic then f is one-to-one <==> ker f = {e}
1.6.1.1.1. الاتجاه الاول سهل وبسيط بثبت فيه ان الكيرنال مفيهوش غير الايدينتيتي
1.6.1.1.2. الاتجاه التاني بثبت انها هومو الاول وارجع اثبت انها وان تو وان
1.6.2. Cayley's Theory proof
1.6.2.1. Let A(G) be the class of all permutations
1.6.2.2. for each a ∈ G define fa: G --> G fa (x) = a . x
1.6.2.2.1. we prove f is permutation (bijective function)
1.6.2.3. H is the set of Permutations we prove that it's a subgroup of G
1.6.2.3.1. closed
1.6.2.3.2. inverse
1.6.2.4. we define another function ψ from G --> H we prove that it's isomorphism
1.6.2.4.1. homomorphism
1.6.2.4.2. bijective
1.7. lec7
1.7.1. theorem
1.7.1.1. any infinite group is isomorphic on (z, +)
1.7.1.1.1. we define a functiom from G --> (z, +) : f (g ^n) = n
1.7.1.1.2. we prove injective (1-1)
1.7.1.1.3. we prove surjective (onto)
1.7.1.1.4. we prove homomorphism
1.8. lec8
1.8.1. every field is integral domain
1.8.1.1. هقول انه اصلا هو commutative ring
1.8.1.2. ,واثبت ان لو في عنصرين ضربهم ف بعض يساوي صفر لازم يكون احدهم بصفر (اثبات ال zero divisor )
1.8.2. in every Ring the following is hold
1.8.2.1. for each a : a.0 = 0.a
1.8.2.2. (a)(-b) = (-a) (b) = -(a.b)
1.8.2.3. (-a) (-b) = (a. b)
1.8.3. in every integral domain the cancellation law is hold
1.8.3.1. بفرض انها انتيجرل دومين واطرح (a .c-)
1.8.4. Theorem
1.8.4.1. subring
1.8.4.1.1. ∀ a, b ∈ S: a-b ∈ S
1.8.4.1.2. ∀ a, b ∈ S: a.b ∈ S
1.8.5. Th: the intersection of two subrings is a subring
1.8.5.1. بفرض عنصر بينتمي لاول سب رينج وبينتمي لتاني سب رينج يبقي بينتمي للتقاطع واحقق الشروط
1.8.6. the intersection of two Ideals is also an ideal
1.9. lec9
1.9.1. Theorem
1.9.1.1. every division Ring is a simple Ring
1.9.1.1.1. هفرض ان في ideal A وعنصر بينتمي ليه لا يساوي الصفر
1.9.1.1.2. هستخدم شرط ال division Ring تبع تواجد عنصر ك انفيرس
1.9.1.1.3. هفرض ان في عنصر بينتمي للرينج واضربه ف a
1.9.1.1.4. وبكده A is not a proper ideal ideal
1.9.1.1.5. R is simple ring since it has no proper ideals
2. definations
2.1. Lec1
2.1.1. def group
2.1.1.1. let G be a non-empty set, defines an opreation *, then the mathematical system is called a group if the following is hold
2.1.1.1.1. closed
2.1.1.1.2. associative law
2.1.1.1.3. identity law
2.1.1.1.4. inverse law
2.1.1.1.5. Abliean (commutative) Group
2.2. lec 2
2.2.1. def order of element | a |
2.2.1.1. let G be a group, a an element ∈ G ≠ e , then the order of a is the least positive integer such that a ^ n = e
2.2.2. def cyclic group
2.2.2.1. let (G, .) be a group,g ∈G it's called cyclic group if and only if G = <g>= {g ^ n : n ∈ z} - (G, + ) be cyclic if G = <g>= {n.g : n ∈ z}
2.2.3. def subgroup
2.2.3.1. let H be a non-empty subset of (G, *) it 's called subgroup if itself forms a group
2.2.4. def Right co-set and Left co-set
2.2.4.1. Let G be a group and H be a subgroup of G , g ∈ G then H =
2.2.4.1.1. Right co-set
2.2.4.1.2. left co-set
2.3. lec 3
2.3.1. def Lagrange Theorem
2.3.1.1. The order of any subgroup of a finite group divides the order of it's Group. i'e |H| | |G| or |G| = t |H| where t is a positive integer
2.3.2. def Normal Subgroup
2.3.2.1. a subgroup H of (G, *) is called normal subgroup if g ^-1 * h * g ∈ H, H * g = g * H
2.3.3. def index
2.3.3.1. the number of distinct (left - right ) co-sets
2.4. lec4
2.4.1. def Center of Group
2.4.1.1. the z(G) center of Group = {x : x .g = g .x ∀ g ∈ G }
2.4.2. def Normalizer
2.4.2.1. Let (G, *) be a group, the normalizer of element a ∈ G defined by N(a) = { x: x * a = a * x } = { e. a }
2.4.3. def Factor group
2.4.3.1. Let (G , * ) be a group it's called factor group if N ⊲ G : G / N = { N * g : ∀ g ∈ G}
2.4.4. def Simple Group
2.4.4.1. let (G, * ) be a group, it' s called simple group if it hasn't any proper normal subgroups
2.4.4.2. any group ( Zp , +p) where p is prime, forms a simple group
2.5. lec5
2.5.1. def homomorphism
2.5.1.1. let F: (G, *) b --> (G`, *) be a mapping group from a group (G, *) to a group (G`, *`), then f is called homomorphism if ∀ a, b ∈ G : f(a *b) = F(a) *` F(b)
2.5.1.1.1. Monomorphism
2.5.1.1.2. Epomorphism
2.5.1.1.3. Endomorphism
2.5.1.1.4. isomorphism
2.5.1.1.5. automorphism
2.5.2. def Kernel of homomorphism:
2.5.2.1. let F: (G, *) b --> (G`, *) be homomorphism then we define the ker f = { x ∈ G: f (x) = e` }
2.6. lec6
2.6.1. Def Cayley's Theorem
2.6.1.1. every finite group is isomorphic to a permutation group
2.7. lec7
2.7.1. Commutator (drived) Group
2.7.1.1. [a, b] = a^-1 . b ^1 . a . b where a , b ∈ G
2.7.1.2. any abelian Group its Commutator has the identity
2.7.2. def Rings
2.7.2.1. A mathematical system (R, +, .) is called a Ring if
2.7.2.1.1. (R, + ) Abelian Group
2.7.2.1.2. (R, .) semi Group
2.7.2.1.3. distributive law
2.7.3. def commutative Ring
2.7.3.1. a Ring (R, +, .) is called commutative ring if
2.7.3.1.1. ∀ a, b ∈ R : a.b = b.a
2.7.4. def Ring with unity
2.7.4.1. a Ring (R, +, .) is called a Ring with unity if it has identity element
2.7.5. def Unity
2.7.5.1. an element x ∈ (R, +, .) is called unity if x ^ -1 exists
2.7.6. def Zero divisor
2.7.6.1. in a Ring (R, + , . ) , a ∈ R ≠ Ø, is called a zero divisor if there exists b ≠ 0 : a.b = 0
2.7.7. def Integral Domain
2.7.7.1. A Ring (D, +, .) is called integral domain if
2.7.7.1.1. commutative
2.7.7.1.2. and hasn't any zero divisors
2.7.7.1.3. (Zp , +p, .p)
2.8. lec8
2.8.1. def Division Ring
2.8.1.1. A Ring (R, +, . ) is called a division Ring if it's all non zero elements are invertible with the second opreation
2.8.1.1.1. (Z, +, .) is not a division Ring
2.8.2. def field
2.8.2.1. A mathematical System (F, +, .) is called a field if
2.8.2.1.1. (f, + ) forms abelian group
2.8.2.1.2. (f - {0} , .) forms abelian group
2.8.2.1.3. distributive law
2.8.3. def Subring
2.8.3.1. let S be non-empty subset of (R, +, . ) it's called subring if itself forms a Ring
2.8.4. def Ideal
2.8.4.1. a non-empty subset I of a Ring (R, +, .) is called an Ideal if
2.8.4.1.1. ∀ a, b ∈ I : a-b ∈ I
2.8.4.1.2. ∀ a ∈ I, r ∈ R : a. r ∈ I
2.8.4.1.3. ∀ a ∈ I, r ∈ R : r. a∈ I
2.8.5. def Simple Ring
2.8.5.1. a Ring is called simple ring if it hasn't any proper ideals
2.9. lec9
2.9.1. def simple Ring
2.9.1.1. let (R, +, .) be a Ring it's called simple ring if
2.9.1.1.1. it hasn't any proper ideals
2.9.1.1.2. ∀ a ∈ R : a ≠ 0 ∃ b : a.b ≠ 0
2.9.2. def Char of Ring
2.9.2.1. let R be a ring then the least positive integer n is called the char of R if
2.9.2.1.1. ∀ r ∈ R : n.r = 0