Some Conclusions Of The Extension Of Finite Groups

Suppose G is a group with the given length of principal series, this paperstudies the construction of G , with the theory of the extension of groups. Butbecause the isomorphic classification of the finite groups is very huge andcomplicated, and the finite group includes p - groups, nilpotent groups, groupsbetween nilpotent and solvable, unsolvable groups and so on. Moreover the problemof non-commutative simple finite groups has already been solved, this paper juststudies the construction of groups, the principal series length of which is 2 or 3, suchas p - nilpotent groups with principal series length of 2, nilpotent groups withprincipal series length of 2 or 3, and supersolvable groups with the principal serieslength of 2 or 3, then gets some mea ningful conclusions.Theorem3-1-1 Let G be an Abelian group with the principal series length of 2,then G is . pq ZTheorem3-1-2 Let G be a group with the principal series length of2, G′≠1,G′′=1 , then G is a group formed by an elementary Abelian p - groupextended by . q ZTheorem3-1-4 Let G be an unsolvable group with the principal series length of2, then G is a group formed by a norma l subgroup A extended by , and is p Z Aa direct product of some isomorphic non-commutative simple groups.Theorem3-1-6 Let G be a group with the principal series length of 2, and itsprincipal factors are all isomorphic to ( 5), then . n A n≥n n G - A×ATheorem3-1-7 Let G be a group with the principal series length of 2, G′= G , let be the norma l Hall subgroup, then is a group formed by elementarN G yAbelian p - group ( p is an odd prime) extended by non-commutative group witheven order.Theorem3-2-1 Let G be a p - nilpotent group with the principal series lengthof 2, let N -G , ( ) , and . Then is a group formed p P∈Sly G N∩P = 1,G = NP Gby N extended by , | , and is a direct product of some isomorphic p Z p / N Nsimple groups.Deduction3-2-3 Let G be a solvable p - nilpotent group with the principalseries length of 2, then G is a group formed by an elementary Abelian p - groupextended by ( are primes, ). q Z p,q p≠qTheorem3-3-1 Let G be a nilpotent group with the principal series length of 2,then:(1)G is ( , are primes, ; pq Z p q p≠q )(2) G = p2 , and its invariants are (p 2 ) or (p, p ) .Theorem3-4-1 Let G be a nilpotent group with the principal series length of 3,then:(1) G = p 3 ( p is a prime ):its invariants are (p3 ) ,or(p 2 , p) , or (p, p, p) ;let p = 2, then G = -a,b-,a 4 =b 2 = 1,b -1ab =a 3 ;let p = 2, then G = -a,b-,a 4 = 1,b 2 =a 2,b - 1ab =a 3 ;let p≠2, then ; G = -a,b-,a p 2 =b p = 1,b -1ab = a1+ plet p≠2, then G = -a,b-,a p = bp = cp = 1,[a, b]= c,[a, c]= [b, c]= 1 .(2) ( are primes, ); p 2q Z p ,q p≠q(3) ( are primes, ); pq p Z×Z p ,q p≠q(4) ( are primes, ). p q r Z p,q,r p≠q≠r TheoreTheorem3-5-1 Let G be a supersolvable group with the principal series lengthof 2, then(1) G = p 2 , and its invariants are (p 2 ) or (p, p );(2) ; pq G = Z(3) G a,b , ap bq 1, b 1ab ar , rq 1(mod p), r 1(mod p), q ( p 1) . = - - = = - =≡≡/ -Theorem3-6-1 Let G be a supersolvable group with the principal series lengthof 3, then(1) G = p3 ( p is a prime):its invariants are (p3 ) ,or(p 2 , p) ,or (p, p, p) ;let p = 2, then G = -a,b-,a 4 =b 2 = 1,b -1ab =a 3 ;let p = 2, then G = -a,b-,a 4 = 1,b 2 =a 2,b - 1ab =a 3 ;let p≠2, then ; G = -a,b-,a p 2 =b p = 1,b -1ab = a1+ plet p≠2, then G = -a,b-,a p = bp = cp = 1,[a, b]= c,[a, c]= [b, c]= 1 .(2) G is a group formed by norma l subgroup A extended by ,and q ZA = p 2 :(I) let p = 2 , , 4q G = Zor G = {a ,b ,g },a2 = b2 = gq = 1 = [a ,b ]= [a ,g ]= [b ,g ];(II) let q = 2 , G = 2p 2 :i) ; G = {a},a 2p 2 = 1ii) G = {a}×{b}×{c},a p =b p =c 2 = 1,[a ,b ] = [b ,c ] =[c ,a ] =1;iii ) ; G = {a ,b},a p 2 = 1= b2 ,b-1ab = a -1iv) G = {a ,b ,c},a p = b p = c 2 = [a ,b ]= 1,c-1ac = a-1 ,c-1bc = b-1 ;v) G = {a ,b ,c},a p = b p = c 2 = [a ,b ]= 1,c-1ac = a,c-1bc = b-1 .(III) let p ,q be odd primes, p≠q, p < q , ; 2 G = {a},a p q = 1 (IV) let be odd primes, ,then is a group formed p ,q p≠q , p > q G byA extended by , and . q Z A = p 2(3) G = {a,b},a pq = 1,b p = at ,b-1ab = a r ,r ,t∈Z ,r p≡1(mod pq ) ( p ,q areprimes, p≠q ) and t(r -1)≡0(mod pq) ;(4) G = {a ,b},a pq = 1,br = at ,b-1ab = a s ,r ,s∈Z ,s r≡1(modpq ) ( p,q,r areprimes, p≠q≠r ), and t(s -1)≡0(mod pq) ;(5) G is a group formed by the norma l subgroup A extended by , and p Z Ais a non-cyclic group , A = pq ( p ,q are primes, p≠q );(6) G is a group formed by the norma l subgroup A extended by , and r Z Ais a non-cyclic group, A = pq ( p,q,r are primes, p≠q≠r ).