Group Theory - Discrete Mathematics.pptx

Group theory is a branch of abstract algebra that studies groups, which are mathematical structures used to model symmetry and many other concepts in mathematics and science. A group consists of a set of elements and an operation that combines two elements to produce another element in the set

1. Group Theory M. Gayathri, M.Sc., M.Phil. Assistant Professor Department of Mathematics Sri Sarada Niketan college of Science for Women , Karur-5

2. Group A group is a set G equipped with a binary operation ∗ that satisfies the following four properties: Closure : For all elements a,b ∈G , the result of the operation a∗ b is also an element of G . Associativity : For all elements a,b,c ∈G , the operation satisfies ( a ∗b )∗c=a∗( b∗c ). Identity element : There exists an element e ∈G such that for every element a ∗e = e∗a =a. Inverse element : For each element a ∈G , there exists an element a −1∈G such that = =e , where e is the identity element.  

3. Examples of Groups 1 . The integers under addition (Z ,+) Set : Z (the set of all integers). Operation : Addition +. 2. The non-zero real numbers under multiplication (R,×) Set : R(the set of all non-zero real numbers). Operation : Multiplication ×. 3. Symmetric group on three elements S3 ​ Set : The set of all permutations of three elements, say { 1,2,3} Operation : Composition of permutations 4. The set of symmetries of a square (dihedral group D4​ ) Set : The set of all symmetries of a square, including rotations and reflections. Operation : Composition of symmetries.

4. Subgroup A subgroup is a subset of a group that is itself a group, under the same operation as the original group. More formally, if G is a group and H is a subset of G , then H is a subgroup of G if: Closure : For all elements h1 ​,h2​∈H, the result of the operation h1 ​∗h2​ is also an element of H . Identity : The identity element of G , denoted e , is in H . Inverses : For every element h ∈H , its inverse is also in H . If these conditions hold, then H is a subgroup of G , and we write H ≤G (i.e., H is a subgroup of G ).  

5. Homomorphism A homomorphism is a map (or function) between two groups that preserves the group operation. Specifically, if G and H are groups with operations ∗ and ⋅, respectively, a map φ:G →H is a homomorphism if for all elements a,b ∈G , the following condition holds: φ( a ∗b )=φ(a)⋅φ(b ) In other words, the homomorphism φ respects the group structure, meaning that the image of the product of two elements under φ is equal to the product of their images.

6. Properties of a Homomorphism Identity preservation : A homomorphism φ: G→H maps the identity element of G to the identity element of H . Specifically, if and are the identity elements of G and H , respectively, then: φ( )= Inverse preservation : A homomorphism preserves inverses. That is, if a ∈G has an inverse , then the image under φ is the inverse of φ( a) in H . Specifically, for all a ∈G : φ ( )= ( φ  

7. Kernel and Image of a Homomorphism For a homomorphism φ: G→H, the following concepts are important: Kernel : The kernel of φ , denoted ker ( φ), is the set of elements in G that are mapped to the identity element of H : ker ( φ)={ g∈G ∣ φ( g )= ​}. The kernel is always a subgroup of G . Image : The image φ , denoted Im ( φ), is the set of elements in H that are the image of some element in G : Im ( φ)={φ( g)∣ g∈G }. The image is a subgroup of H .  

8. Isomorphism A homomorphism φ:G →H is called an isomorphism if it is bijective (one-to-one and onto). If there exists an isomorphism between two groups, then the groups are said to be isomorphic , meaning they are structurally identical in terms of group theory, even if they are represented differently.

9. Cosets In group theory, cosets are subsets of a group that arise from the action of a group on one of its subgroups. More specifically, given a group G and a subgroup H of G , a coset is formed by taking an element g∈ G and combining it with every element of H using the group operation . Left Coset Given a subgroup H of a group G and an element g∈ G , the left coset of H with respect to g is the set of elements obtained by multiplying g on the left by every element of H . It is denoted by: gH ={ g∗h∣h∈H } Where : gH is the left coset of H with respect to g , ∗ is the group operation in G , h ranges over all elements of H . Right Coset Similarly, the right coset of H with respect to g is the set of elements obtained by multiplying g on the right by every element of H . It is denoted by: Hg ={ h∗g∣h∈H } Where : Hg is the right coset of H with respect to g , h ranges over all elements of H .

10. Normal Subgroup A normal subgroup (or invariant subgroup ) is a special type of subgroup that is "invariant" under conjugation by any element of the group. More formally, a subgroup H of a group G is normal if for every element g ∈G and every element h ∈H , the conjugate of h by g (denoted ) is still an element of H . In other words, H is normal in G if:   all g ∈G . This condition means that the set of conjugates of the elements of H remains within H , and H behaves "symmetrically" in the group G .