A group is solvable iff G^(k)={e} || Characterization Theorem || Solvable Group || M.Sc Maths Sem 1

A group is solvable iff G^(k)={e} || Characterization Theorem || Solvable Group || M.Sc Maths Sem 1

#abstractalgebra #modernalgebra #mscmathematics #grouptheory #highermathematics #algebra #solvablegroup

But before this above theoem we have to cover these theorem which is very very important to learn above theorem.
   • Commutator Subgroup & Theorem on Comm...  
In this video if have covered topic
1. Commutator Subgroup
2. Theorem based on Commutator Subgroup
Theorem : If G' is commutator subgroup of G. Then
(i) G' is normal in G.
(ii) G/G' is abelian.

Theorem : If N is any normal subgroup of G such that G/N is abelian. Then definately Commutator subgroup G' will contain in normal subgroup N. i.e., N is Super Set of Commutator subgroup G'.

Theorem : If any subgroup H of G containing Commutator subgroup G'. Then definately H will be a normal subgroup of G.

Solvable Group and Solvable series and their conditions
   • Definition of Solvable Group || Solva...  

More Important Theorems for M.Sc Sem 1 (Modern Abstract Algebra)
(1) First Law of Isomorphism/Fundamental Theorem on Homomorphis
   • First Law Of Isomorphism OR Fundament...  

(2) Second Law of Isomorphism
   • Second Law of Isomorphism Theorem || ...  

(3) Third Law of Isomorphism
   • Third Law Of Isomorphism Theorem || B...  

(4) Jordan Holder Theorem
   • Jordan Holder Theorem (Complete Proof...  

(5) Every subgroup of a solvable group is solvable
   • Every subgroup of a solvable group is...  

(6) Upper Central Series, Lower Central Series and Nilpotent Group
   • Upper Central Series, Lower Central S...