Microsoft word mal 511
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Microsoft word mal 511
MAL-511: M. Sc. Mathematics (Algebra)Lesson No. 1Lesson: Subnormal and Normal series-1Written by Dr. Pankaj Kumar N etted by Dr. Nawneet HoodaSTRUCTUR Microsoft word mal 511 RE.1.0 OBJECTIVE.1.1INTRODUCTION.1.2SUBNORMAL AND NORMAL SERIES1.3ZASSENHAUS LEMMA AND SCHREIER’S REFINEMENT THEOREM.1.4COMPOSITION SERIES.1.5COMMUTATOR SI BGROUP.1.6MORE RESULTS ON COMMUTATOR SITĨGROĨTS.1.71NVA1UANT SERIES AND CHIEF SERIES.1.8KEYWORDS.1.9SUMMARY.1.10SELF ASSESMENT QUESTIONS.1.11SUG Microsoft word mal 511 GESTED READINGS.1.0 OBJECTIVE. Objective of this Chapter is to study some properties of groups by studying the properties of the senes of its subgroupMicrosoft word mal 511
s and factor groups.1.1INTRODUCTION, since groups and then subgroups have some relation, therefore, in this Chapter we use subgroups of given group toMAL-511: M. Sc. Mathematics (Algebra)Lesson No. 1Lesson: Subnormal and Normal series-1Written by Dr. Pankaj Kumar N etted by Dr. Nawneet HoodaSTRUCTUR Microsoft word mal 511 eries, commutator subgroups and then properties and three subgroup lemma of p. Hall. In Section 1.2. we study subnormal and normal series. It is also shown that every normal senes is a subnormal but converse may not be true. In Section 1.3, we study Zassenhaus Lemma and Schreier’s refinement theorem Microsoft word mal 511 . In Section 1.4, we study composition series and see that an abelian group has composition series if and only if It IS finite. We also study Jordan HMicrosoft word mal 511
older theorem which say that any two composition series of a finite group are1equivalent At the end of this chapter we study some more series namely CMAL-511: M. Sc. Mathematics (Algebra)Lesson No. 1Lesson: Subnormal and Normal series-1Written by Dr. Pankaj Kumar N etted by Dr. Nawneet HoodaSTRUCTUR Microsoft word mal 511 0=Go30i3G2=... ->Gn=(e)of subgroups of G is called subnormal series of G if Gi is a normal subgroup of Gm for each i, lseries of a group). A finite sequenceG=Gợ“>G)“>Gí7)... ~)Ga=(e)of subgroups of G is called normal series of G if each Gi is a normal subgroup of G for Microsoft word mal 511 1Microsoft word mal 511
his is normal as well as subnormal series lor G.1.2.3Theorem. Prove that every normal series of a group G is subnormal but converse may note be true.PMAL-511: M. Sc. Mathematics (Algebra)Lesson No. 1Lesson: Subnormal and Normal series-1Written by Dr. Pankaj Kumar N etted by Dr. Nawneet HoodaSTRUCTUR Microsoft word mal 511 and for even g "G. we have (gi)’1 g giCGi- Since Gi r Gm <- G. Hence for every gifcGi and for every gi.|fcGi.|, we have (gi-i)’1 gi gi-ifcGi r.c. Gi is normal in Gm. Hence (*) is subnormal series for G also.For converse part take G = s4, symmetric group of degree 4.Then the sequence Microsoft word mal 511 MAL-511: M. Sc. Mathematics (Algebra)Lesson No. 1Lesson: Subnormal and Normal series-1Written by Dr. Pankaj Kumar N etted by Dr. Nawneet HoodaSTRUCTURGọi ngay
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