Maths cafe

  Theorem : Let 𝐻,𝐾 be cyclic groups of order 𝑚, 𝑛 respectively, where 𝑚, 𝑛 are relatively  prime. Then 𝐻 × 𝐾 is a cyclic group of order 𝑚𝑛. Theorem:  Let 𝐺 = 𝐻 × 𝐾 and 𝐻1 be a normal subgroup of 𝐻. Then 𝐻1 is normal in 𝐺. Theorem:  Let 𝐺 = 𝐻 × 𝐾. Then the factor group 𝐺/𝐾≅ 𝐻. Theorem:  let 𝐺 be a group. Then a) the derived group 𝐺′is normal subgroup of 𝐺, b) the factor group 𝐺/𝐺′is abelian, c) if 𝐾 is a normal subgroup of 𝐺 such that 𝐺/𝐾 is abelian then 𝐺′ ⊆ 𝐾. Theorem:  Let 𝜁 (𝐺) be the centre and 𝐼(𝐺) be the inner automorphism of a group 𝐺. Then 𝐺 𝜁( 𝐺 )≅ 𝐼(𝐺). Theorem : The set 𝐼(𝐺) of all inner automorphism of a group 𝐺 is a normal subgroup of  𝐴(𝐺). Theorem:  Let 𝐺 be a group. The mapping 𝜑 ∶ 𝐺 ⟶ 𝐺 defined by𝜑 (𝑔 )= 𝑔−1, 𝑔 ∈ 𝐺 is an automorphism if and only if 𝐺 is abelian. Theorem : The set 𝐴 𝐺 of all automorphism of 𝐺 form a group. Theorem:  Let 𝐺 be a direct product of it...

 Homomorphism: Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be homomorphism if  𝜑 (𝑥 ∙ 𝑦)= 𝜑( 𝑥)∗ 𝜑(𝑦)for 𝑥, 𝑦 ∈ 𝐺. The range of 𝜑 in 𝐻 is called the homomorphic image of 𝜑. Endomorphism:  Let (𝐺 ,∗) be a group. A homomorphism 𝜑 ∶ 𝐺 ⟶ 𝐺 is called endomorphism. Monomorphism: Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be monomorphism if  a) 𝜑 is homomorphism. b) 𝜑 is injective. Epimorphism: Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be  epimorphism if  a) 𝜑 is homomorphism. b) 𝜑 is surjective. i.e., for all 𝑏 ∈ 𝐻, there is an element 𝑎 ∈ 𝐺 such that 𝜑 (𝑎) = 𝑏. Endomorphism Let (𝐺 ,∗) be a group. A homomorphism 𝜑 ∶ 𝐺 ⟶ 𝐺 is called endomorphism.  Isomorphism Let (𝐺 , ∙) and (𝐻 ,∗) be two groups. A mapping 𝜑 ∶ 𝐺 ⟶ 𝐻 is said to be  isomorphism if  a) 𝜑 is homomorphism. b) 𝜑 is injective. c) 𝜑 is surjective. The isomorph...

 Quaternion Group: The quaternion group 𝑄8 is a non-abelian group of order 8, isomorphic to  the certain eight elements subset of the quaternions under multiplication. It is given by 𝑄8 = {±1, ±𝑖, ±𝑗, ±𝑘}. Where 𝑖^2 = 𝑗^2 = 𝑘^2 = −1, and 𝑖.𝑗 = 𝑘 = −𝑗. 𝑖 𝑗. 𝑘 = 𝑖 = −𝑘.𝑗 𝑘. 𝑖 = 𝑗 = −𝑖. 𝑘. Since 𝑖.𝑗 ≠ 𝑗. 𝑖, therefore it is non-abelian. There are 6 subgroups of 𝑄8 of order 1,2,4 and 8. These are  𝐻1 = 1 , 𝐻4 = {±1, ±𝑗 }  𝐻2 ={1, −1} , 𝐻5 ={ ±1, ±𝑘 }  𝐻3 = {±1, ±𝑖} , 𝐻6 ={ ±1, ±𝑖, ±𝑗, ±𝑘} = 𝑄8 All these subgroups are cyclic and abelian.  Properties: a) The quaternion group 𝑄8 has the same order as the dihedral group  𝐷4 =< 𝑎, 𝑏: 𝑎^4 = 𝑏^2 = (𝑎𝑏)^2 = 1 >. b) Every subgroup of 𝑄8 is a normal subgroup. c) The center and the commutator subgroup of 𝑄8 is the subgroup {1, −1}. d) The factor group 𝑄8/{1,−1} is isomorphic to the Klien four group 𝐾4. Q8 is solvable group. Q8 is nilpotent group Q8 meta ablian...

 Direct Product: If 𝐺 and 𝐻 are two groups (finite or infinite). Then the direct product of 𝐺 and 𝐻 is a  new group, denoted by 𝐺 × 𝐻 and is defined by 𝐺 × 𝐻 = 𝑥, 𝑦 𝑥 ∈ 𝐺, 𝑦 ∈ 𝐻}. The group operation defined is multiplication. Let 𝑎, 𝑏 ∈ 𝐺 and 𝑥, 𝑦 ∈ 𝐻, then 𝑎, 𝑏 . 𝑥, 𝑦 = (𝑎. 𝑥, 𝑏. 𝑦). It is also called the external direct product. Properties: a) Identity: The direct product 𝐺 × 𝐻 has an identity element, namely {𝑒1, 𝑒2}, where 𝑒1 ∈ 𝐺 and 𝑒2 ∈ 𝐻. b) Inverse: The inverse of each element (𝑥, 𝑦) ∈ 𝐺 × 𝐻 is (𝑥−1, 𝑦−1), where 𝑥−1 ∈ 𝐺 and 𝑦−1 ∈ 𝐻. c) Associativity:  The associative law holds in 𝐺 × 𝐻. That is, for 𝑥1, 𝑦1 , 𝑥2, 𝑦2 , (𝑥3, 𝑦3) ∈ 𝐺 × 𝐻 ( 𝑥1, 𝑦1 . 𝑥2, 𝑦2). 𝑥3, 𝑦3= 𝑥1, 𝑦1. ( 𝑥2, 𝑦2.(𝑥3,𝑦3)).

 Complete Group:  if the centre 𝜁( G) of a group 𝐺 is trevial and very automorphism of 𝐺 is an inner  automorphism, 𝐺 is called a complete group Derived Group OR Commutator Subgroup:  Let 𝐺 be a group and 𝐺′ be a subgroup of 𝐺. Then 𝐺′is said to be a commutator subgroup, if it is generated by a set of commutators of 𝐺. Simple group :  A group is said to be simple group if it has no proper normal subgroups.

 First Isomorphism Theorem Let 𝜑: 𝐺 ⟶ 𝐺′ be an epimorphism from 𝐺 to 𝐺′. Then: a) The 𝐾 = 𝑘𝑒𝑟𝜑 is a normal subgroup of 𝐺. b) The factor group 𝐺𝐾is isomorphic to 𝐺′. c) A subgroup 𝐻′ of 𝐺′is normal in 𝐺′ if and only if its inverse image 𝐻 = 𝜑−1(𝐻′) is normal in 𝐺. d) There is one-one correspondence between the subgroups of 𝐺′ and those subgroups of 𝐺 which contain 𝑘𝑒𝑟𝜑 Second Isomorphism Theorem Let 𝐻 be a subgroup and 𝐾 be a normal subgroup of a group 𝐺, then; a) 𝐻𝐾 is a subgroup of 𝐺, b) 𝐻 ∩ 𝐾 is normal in 𝐻, and c) 𝐻𝐾/𝐾≅𝐻/𝐻∩𝐾 Third Isomorphism Theorem Let 𝐻,𝐾 be normal subgroups of a group 𝐺 and 𝐻 ⊆ 𝐾. Then  (𝐺 𝐻) /(𝐾 𝐻) ≅ 𝐺 /𝐾.

 Normalizer of a Subgroup:  Let The set of those elements of which permute with is called normalizer of in Mathematically, it can be written as:   NG(H)={gH=Hg : g∈G} Results: 1. Normalizer of a subgroup of a group is always a subgroup of    2. In an abelian group   subgroup of  3. For any subgroup of H of G Z(G) ≤NG(H)   4. Let H ≤G and |H |= 2  then CG(H)=NG(H)     5. Let G  be a finite group then a∈Z(G) and a∈NG(H)   6. Let H ≤ G | G| =p^n if p divides [G:H] then NG(H) ≠H    7. For any a ∈ G , Z(G)= ⋂a ∈G NG(a)     8. For any group  ⟨e ⟩ = G=NG(G)   9. For any group G NG(Z (G) ) =G

 Centralizer of a Subgroup:  Let be a subgroup of The set of those elements of which commute with every element  of is called centralizer of in Mathematically, it is written as:{g∈G:gh=hg for all h∈H}   Note: Centralizer can also be defined for any subset of a group  1 CG(H) is non-empty 2 CG (H) is a subgroup of G for any subgroup H of G 3 if G be an ablian group CG(H)=G 4 The centerlizer of a subgroup need not contain that sub group 5 The Centralizer of the whole group is the center of the group 6 The number of element in a conjugacy class Ca of an element a∈G is equal to Index of it's normalizer in G thus |Ca|=|G:CG(a)|.

 Def. Klein’s four group : A group of order four in which every element is self-inverse or every non-identity  element is of order 2 is called Klein’s four group. Symbolically, V4={a, b, c, d} such that ab=ba=c bc=cb=a ca=ac=b and a^2=b^2=c^2 =e Properties: a) Every non-identity element is of order 2. b) Any two of the three non-identity element generates the third one. c) It is the smallest non-cyclic group. d) All proper subgroups of V4 are cyclic. Subgroups of V4: H1=e 𝐻2 ={ 𝑒, 𝑎} ,  𝐻4 ={ 𝑒, 𝑐 }  𝐻3 = {𝑒, 𝑏},  𝐻5 = 𝐾4

 Centre of a group :  Let G be a group then centre of the group G is defined to be subset of all elements of G which commute with every element of G and it is denoted by Z(G). In symbols, Z (G )={a ∈G, ax= xa for all x in G  Results : 1. Z (G) is a subgroup of G. 2. G is abelian iff G= Z (G ). 3. Let G be a group and a ∈Z( G) then in the composition table of G the row and column headed by ‘a’ are same. 4. As zn and  U (n) are abelian groups so they are centres of themselves i.e., z(zn)=zn and  Z(U(n))=U(n). 5 The Center of a group is ablian part of the group. 6 The Center of any group is non empty 7 The Center z(G) of a group G is a subgroup of G. 8 The Center of a group G is normal subgroup 9 The Center of a group is characteristics subgroup. 10 The Center of a group may not fully invariant subgroup. 11 The Center of an ablian group is the whole group and therefore cyclic group also.

 Def. Cyclic group :  A group G is said to be cyclic if there exists an element aG such that every elementof G is of the form a^n, where n is an integer. The element ‘a’ is then , called a generator of G and we write : G = < a > or (a)  Results on cyclic groups :  1. Every cyclic group is abelian. 2. Every subgroup of a cyclic group is cyclic. 3. If ‘a’ is a generator of a cyclic group G, then a^-1is also a generator of G. 4. The order of a cyclic group is equal to the order of its generator. 5. If a finite group of order n contains an element of order n then the group must be cyclic. 6. Every group of prime order is cyclic . 7. Every infinite cyclic group has exactly two generators. 8. The number of generators of a finite cyclic group of order n is. Ø(n ), where Ø denotes the Euler’s Ø function. 9. Let G be a finite cyclic group such that o(G)= n and G=<a> then a^m will be the generator of G if and only if gcd(n, m) 10. A group of prime order has no pro...

 Examples of sub groups: i. (ℤ, +) is a subgroup of ℚ , + and ℚ , + is a subgroup of ℝ , + . ii. The set ℚ+ under multiplication is a subgroup of ℝ+ under the algebraic operation multiplication. iii. The 𝑛𝑡h root of unity in ℂ𝑛 form a subgroup 𝑈𝑛 of the group ℂ∗ of non-zero complex numbers under the algebraic operation multiplication.  Theorem :   A non-empty subset 𝐻 of a group 𝐺 is a subgroup of 𝐺 if and only if for any pair  of 𝑎, 𝑏 ∈ 𝐻 , 𝑎𝑏−1 ∈ 𝐻 ; 𝑎 ≠ 𝑏 ≠ 𝑒

   Definition : Subgroup :  A non empty subset H of a group G is called a subgroup of G , if H itself is a groupw.r.t.the same binary operation as in G. Def. Proper and Improper subgroups :  For a group G, the set {e} and G are always subgroups of G and are called improper subgroups of G.  Any other subgroup [other than G and {e}] is called a proper subgroup

Involution:  An element 𝑥 of order 2 in a group 𝐺 is called an involution.

 Order of a Group The number of elements in a group is called the order of a group and is denoted by |G|. Order of an element Let 𝑎 be any element of a group G. A non-zero positive integer 𝑛 is called the order of 𝑎 if 𝑎^𝑛 = 𝑒 and 𝑛 is the least such integer. Periodic Group A group all of whose elements are of finite order is called a periodic group. A finite group is periodic

 Historical Note about abelian group or  Commutative Commutative groups are called abelian in honor of the Norwegian mathematician Niels Henrik Abel (1802 − 1829). Abel was interested in the question of solvability of polynomial equations. In a paper written in 1828, he proved that if all the roots of such an equation can be expressed as rational functions 𝑓, 𝑔, … , h of one of them, say 𝑥, and if for any two of these roots, 𝑓(𝑥) and 𝑔(𝑥), the relation 𝑓 𝑔 𝑥 = 𝑔 𝑓 𝑥 always holds, then the equation is solvable by radicals. Abel showed that each of these functions in fact permutes the roots of the equation; hence, these functions are elements of the group of permutations of the roots. It was this property of commutativity in these permutation groups associated with solvable equations that led Camille Jordan in his 1870 treatise on algebra to name such groups abeli;the name since then has been applied to commutative group in general.

 Torsion Free And Mixed Group: A group in which every element except the identity element 𝑒 has infinite order is  known as torsion free (𝑎-periodic or locally infinite). A group having elements both of finite as well as  infinite order is called a mixed group. Semigroup And Monoid: A set with an associative binary operation is called a semigroup.  A semigroup that has an identity element for the binary operation is called monoid. Note that every group is both a semigroup and a monoid. Abelian Group: A group 𝐺 is abelian if its binary operation is commutative. That is,let (𝐺, ∗) be a  group. Let , 𝑏 ∈ 𝐺 , then 𝐺 is called an abelian group iff  𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 Examples a. The familiar additive properties of integers and of rationals, real and complex numbers show that ℤ, ℚ, ℝ and ℂ under addition abelian groups. b. The set ℤ+ under addition is not a group. There is no identity element for + in ℤ+. c. The set ℤ+ under multiplication is not a group. T...

 Historical Note: There are three historical roots of the development of abstract group theory evident in the mathematical literature of the nineteenth century: the theory of algebraic equations, number theory and geometry. All three of these areas used group theoretic methods of reasoning, although the methods were considerably more explicit in the first area than in the two.                     One of the central themes of geometry in the nineteenth century was the search of invariants under various types geometric transformations. Gradually attention became focused on the transformations themselves, which in many cases can be thought of as elements of groups.        In number theory, already in the eighteenth century Leonhard Euler had considered the remainders on division of power 𝑎𝑛 by fixed prime 𝑝. These remainders have “group” properties. Similarly, Carl F. Gauss,  Finally, the theory of algebraic...

Properties of group (A) cancelation law hold in G that is a*b=b*c  Implies b=c for all a, b, c,belong to G (B) identity element is unique (C) inverse of an element is unique in G

Groups A pair (G, *) where G is a non- empty set * a binary operation in G is a group if and only if (1) The binary operation * closed a *b=b* a a, b belong to G (2) The binary operation * is associative   (a*b)*c= a*(b*c) (3) There is an identity element e belong to G, such that for all a belong to G a* e= e* a=a (4) For each a belong to G there is element a`  belong to G such that a* a`=a`*a= e where a` is called the inverse of a in G