Finite Groups Whose Third Or Fourth Maximal Subgroups Are All Cyclic

The properties of the n-th maximal subgroup ha^s a great influence on the structure of groups.In this thesis,we study finite groups in which all fourth maximal subgroups are cyclic.Let G be a finite nontrival group,l(G)denotes the number m of G,satisfied that all m-maximal subgroups are cyclic of G but there exists a(m-1)-maximal non-cyclic subgroups.Let n be positive integer and p₁^?₁p₂^?₂…p_k^?_k its standard factorization,where p11+?₂+…+?_k,?(n)=k and?(1)=0.Set P(n)=0 if ?(n)=0 or 1,Set P(n)=min{?₁,?₂,…,?_k} if?(n)>1.Let Q₂ⁿ be the generalized quaternion group of order 2ⁿ,Zn the cyclic group of order n,F_qr a Frobenius group of order qr with kernel Z_q and complement Z_r and Dn the dihedral group of order n,SmallGroup(n,i)the i small group of nth order groups in GAP software.We prove the following results.Theorem A Let G be a finite group and p,q,r distinct primes.Then l(G)=3 if and only if G is isomorphic to one of the following groups:(1)G is a group of order p~4 except Z_p~4 and Q₁₆;(2)G?Q₃₂;(3)G=(a,b|a^m=bⁿ=1,b^-1ab=a^s),((s-1)n,m)=1,2 ??(G)?4,Sⁿ?1(mod m),|G|=nm,?(m)+?(n₀)+P(n/n₀)=4,where n₀ is the smallest positive integer such that sno?1(mod m);(4)G?P(?)Z_q,where P is a group of order p~3 except Z_p~3 and Q₈;(5)G?Q₁₆×Z_q,where q is an odd prime;(6)G?Z_q(?)P,where P is Q₁₆ or a group of order p~3 except Z_p~3 and Q₈;(7)G?(Z_p × Z_p)(?)Z_q2,Z_q2(?)(Z_p × Z_p);(8)G?Z_q2(?)Q₈,where q is an odd prime;(9)G?(Z_p×Z_p)(?)(Z_p×Z_q);(10)G?SmallGroup(72,3),SmallGroup(72,25;(11)G?Z_r(?)SL(2,3),where r? 5 is a prime;(12)G?Z_r(?)(Z_q(?)Q₈),where q,r are distinct odd primes;(13)G?F_qr(?)(Z_p × Z_p);(14)G?Z_r(?)(Z_p × Z_p)(?)Z_q);(15)G?Z_r(?)(Z_q(?)(Z_p × Z_p));(16)G?(Z_p×Z_p)(?)Z_qr.where Z_qr acts faithfully on Z_p ×Z_p and qr | p^2-1;(17)G?(Z_p×Z_p)(?)D_2q,where D_2q acts faithfully on Z_p×Z_p and p,q are distinct odd primes;(18)G?SL(2,5);(19)G?PSL(2,p),where p=5,13 or p a prime,?(p-1)=?(p+1)=3,p?±3(mod 8)and p(?)±1(mod 10).Theorem B Let G be a non-solvable.Then l(G)=4 if and only if G is isomorphic to one of the following groups:(1)G?PSL(2,q),where q=7,8,9,11,19,27,29 or q a prime and max{?(q-1),?(q+1)}=4;(2)G?SL(2,p),where p a prime and p=7,11,13,19,29 or ?(p-1)=?(p+1)=3,p ?±3(mod 8),p(?)±1(mod 10)or ?(p?1)?3,?(p+1)=4;(3)G?SmallGroup(240,89);(4)G?SL(2,5)× Z_r,where r is an odd prime;(5)G?PSL(2,p)× Z_r,where p,r are primes,p=5,13 or ?(p-1)=?(p+1)=3,p(?)±3(mod 8)and p(?)±1(mod 10);(6)G?PGL(2,p),where p=5,13 or p a prime,?(p-1)=?(p+1)=3.