# Basics of Group Theory in Mathematics

Introduction to sets in mathematics, concepts of semigroup and monoid, definition of a group, examples of subgroups, cyclic groups, normal subgroups, simple groups, and factor groups. Exploring fundamental principles in group theory.

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**Basics of Group Theory**B.Sc-II Sem III Unit-I By By Mr.M.S.Wavare Mr.M.S.Wavare Department of Mathematics Department of Mathematics Rajarshi Rajarshi Shahu Shahu Mahavidyalaya Mahavidyalaya, , Latur (Autonomous) (Autonomous) Latur**Introduction**Set: The collection of well defined objects is called set. Examples:- A={1,2,3,4,5} N={1,2,3, } W={0,1,2,3, .} I={ .,-3,-2,-1,0,1,2,3, .} R={real numbers}**Semigroup & Monoid**The non empty set G and * be a binary operation on G is said to be semigroup if it satisfied associative property. Example: N={set of natural numbers} is a semi group. A semigroup (G,*) having an identity element is called a monoid. Example: I+={0,1,2,3 }**Definition of Group**iv) Closure: for all a,b in G, a+b G**Cyclic Group**A group G is cyclic if there is an element a in G, such that every element of G is some power of a The group G is said to be generated by a, and a is called as generated of G. Examples: Z is an infinite cyclic group and Z=<1,-1>**Normal Subgroup**A subgroup N of a group G is said to be a normal subgroup of gG if gN=Ng for all g in G. Every subgroup of abelian group is normal. Every subgroup of index 2 is normal.**Simple Group: A simple group is a group of**order greater than 1 whose only normal subgroups are the identity subgroup and group itself. Factor Group: Let H be a normal subgroup of a group G. Then, G/N be the set of all coset of H in G is a group with respect to the binary operation defined by aHbH=abH, for all aH,bH in G/H. It is called as Factor Group.