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