Table of Contents

**MTH633: Group Theory**

In modern algebra, group research is a system that combines a set of objects and binary functions that can be used in two sets of a set, which include specific axioms. MTH633 Handouts pdf

# MTH633 Handouts pdf

**Course Category: Mathematics**

**Course Outline**

*Properties of real and complex numbers, Binary Operation, Bijective maps, Inversion Theorem, Isomorphic binary structures, Groups, Examples of Groups, Uniqueness of identity and inverse, Elementary properties of groups, Abelian Groups, Order of a Group, Finite Groups, Subgroups, Examples of subgroups, Two-Step subgroup test, One Step subgroup test, Cyclic Groups,*

*Permutation Groups, Examples of Permutation Groups, Theorem on permutation groups, Cayley”s Theorem, Orbits, Cycles, Disjoint Cycles, Cycle Decomposition, Parity of permutations, Alternating Group, Direct product, Finitely generated abelian groups, Cosets, Partition of group *

*Lagrange’s Theorem, Applications of Lagrange”s theorem, Indices of subgroups, Converse of Lagrange”s Theorem, Homomorphism of Groups, Properties of homomorphism, Normal Subgroups, Morphism Theorem for groups, Application of Morphism theorem, properties of homomorphism, Normality of kernel of homomorphism, Normal group, Factor group, Cosets multiplication & Normality*

*Examples on a kernel of homomorphisms and group homomorphism, Factor group from homomorphism, Kernel of an injective homomorphism, Factor groups from Normal subgroups, Example of Morphism theorem of groups, Normal groups, and Inner Automorphism, Factor group computations*

*Simple Group, Maximal Normal Subgroups, The Centre subgroup, Example of the Centre subgroup, Commutator subgroup, Generating set, Commutator subgroup, Automorphisms, Group Action on set, Stabilizer, Orbits, Conjugacy and G-sets*

*Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism Theorem, First Sylow Theorems, Second Sylow Theorems, Third Sylow Theorems, Application of Sylow Theorems *

MTH633 Handouts pdf

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### MTH633 HANDOUTS

**MTH633: Group Theory**

#### WHAT IS A GROUP ??

Group research is known as group theory. If there is a limited number of elements, a group is called a limited group and the number of elements is called a group order group. A subset of a closed group under a group function and the opposite function is called a subgroup.

**Cyclic **groups

Among the first mathematical algorithms, we study is the division algorithm of integers. It says that if given the whole number m and the whole number of divider d there is a quotient q and a residual r <d so that m / d = q + r / d. This is very easy to prove and we encourage the reader to do so.

#### Lema. (Division Algorithm)

When whole numbers are given m and d> 0, there are especially determined numbers d and r satisfying m = dq + r and 0 ≤ r <d. By using the division algorithm we can get exciting results about cycle groups.

**Conjugation**

X and y should be part of group G. If there is a g ∈ G equal to g −1xg = y, then we say x plus y. The relationship “x is conjugate to y” is a relationship of equality and equality classes are called conjugacy classes. Specifies the conjugacy category of x by K (x). Thus, K (x) = {g −1xg: g ∈ G}.

If x is part of group G, then it is easy to see that K (x) = {x} if and only if x goes through the whole part of G. So, in particular, conjugacy classes abelian groups are not interesting. . Note that x ∈ G, | K (x) | = 1 if and only if x ∈ Z (G). Next if and only if x ∈ Z (G). Therefore i