Maths cafe

 Cauchy theorem for finite abelian groups :  If G is a finite abelian group such that p/ o( G) , p is a prime number , then there exists an element a( ≠ e) ∈ G such that  a^p= e  i.e. o(a) = p. Cauchy theorem for finite groups :  If G is a finite group such that p/ o (G)  , p is a prime number , then  there exists an element of order p in G.

 Let 𝐺 =< 𝑎, 𝑏: 𝑎^3 = 𝑏^2 = (𝑎𝑏)^2 = 1 > be the dihedral group of order 8. Its  elements are {1, 𝑎, 𝑎^2,  𝑏, 𝑎𝑏, 𝑎^2𝑏, }.  Subgroups of D3: <1>={1}, <a>={a , a^2,a^3=1} <a^2>={a^2,a^4=a, a^6=1} <b>={b, b^2=1} <ab>={ab, 1} <a^2b>={a^2b, 1} Center of D3: Z(D3)={e}.               because n is odd Commutator subgroup of D3: {1,a, a^2} Normal subgroup of D3: t(3)+1=2+1=3 Cyclic subgroup of D3: t(3)+3=2+3=5 D3 smallest non-ablian group  

  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.