II Sem III UnitI By MrMSWavare Department of Mathematics Rajarshi Shahu Mahavidyalaya Latur Autonomous Introduction Set The collection of well defined objects is called set ID: 1025264
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1. Basics of Group TheoryB.Sc-II Sem –IIIUnit-IBy Mr.M.S.WavareDepartment of Mathematics Rajarshi Shahu Mahavidyalaya, Latur(Autonomous)
2. IntroductionSet: 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}
3. Semigroup & MonoidThe 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……}
4. Definition of Group iv) Closure: for all a,b in G, a+bєG
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7. Examples of Subgroups
8. Cyclic GroupA group G is cyclic if there is an element a in G, such that every element of G is some power of aThe 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>
9. Normal SubgroupA 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.
10. 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.