The Influence Of The Number Of Some Special Subgroups On The Structure Of Finite Group

In finite group theory, it is a very important topic to investigate the structure of group through discussing the properties of its subgroups which is also an important starting point to research the structure of groups. For example, many important results were obtained by discussing cyclic subgroups and normal subgroups. With the development of the research, we found that the influence of the number of non-cyclic subgroups or non-normal subgroups on the structure of group is very significant.About the influence of non-cyclic subgroups on the structure of finite group. Let δ(G) be the number of the conjugacy classes of non-cyclic subgroups in G. Obviously δ(G)=0if and only if G is a cyclic group; δ(G)=1if and only if G is a non-cyclic group whose proper subgroups are all cyclic; Shirong Li, Xubo Zhao gave complete classification of finite groups all of whose soluble subgroups H satisfy δ(H)≤2; Wei Meng, Jiakuan Lu obtained the complete classification of finite nilpotent groups with δ(G)=4. About the influence of non-normal subgroups on the structure of finite group. Let v(G) be the number of the conjugacy classes of non-normal subgroups in G. The classification of finite groups with v(G)<4were obtained. Meanwhile, the influence of the number of non-normal subgroup on the structure of finite group was another research direction. Huaguo Shi, Lv Gong gave the structure of finite groups with2,5,7non-normal subgroups. Yuemei Mao obtained the structure of finite group with p conjugate non-normal subgroups.Here we mainly discussed two issues. Firstly, we studied the influence of the number of non-cyclic subgroups on the structure of finite group. First, we simplified the proof of finite nilpotent but not p-groups with δ(G)=4of which the classification was given by Wei Meng, Jiakuan Lu in2009. And on this basis, we continued the study of finite nilpotent group with δ(G)=5and δ(G)=6, thus we obtained the complete classification of finite group with δ(G)≤6. Secondly, we discussed the influence of the number of non-normal subgroups on the structure of finite group. Let τ(G) be the number of non-normal subgroups in G. The author used mainly the complete classifications of finite group having exactly1,2,3,4conjugacy classes of non-normal subgroups to obtain the structure of finite group with τ(G)=11. We obtained:Theorem3.3If G is a finite nilpotent group with δ(G)=5, then G is isomorphic to one of the following groups:(1)<a, b, c|a3=b3=c3=1,[a, b]=c,[a, c]=[b, c]=1>;(2) Zp5×Zp;(3)<a,b|ap5=bp=1,ab=a1+p4);(4) Q32=<a,b|a16=1,b2=a8,ab=a-1>;(5) D16=<a,b|a8=b2=1,ab=a-1>;(6) Zp×Zp×Zq4, where p, q are distinct primes;(7) Q8×Zq4, where q is an odd prime.Theorem3.4If G is a finite nilpotent group with δ(G)=6, then G is isomorphic to one of the following groups:(1) Zp6×Zp;(2)<a,b|ap6=bp=1,ab=a1+p5>;(3) Zp×Zp×Zq5;(4) Zp×Zp×Zq2×Zr;(5) Q8×Zq5;(6) Q8×Zq2×Zτ;(7) Zp2×Zp×Zq2; (8)<a,b|ap2=bp=1,ab=a1+P>×Zq2,p>2；(9)Zp3×Zp×Zq；(10)<a,b|ap3=bp=1,ab=a1+p2>×Zq；(11)Q16×Zq;(12)D8×Zq.Theorem4.6If G is a finite group with τ(G)=11,then G is isomorphic to one of the following groups:(1)<x,y|x11=y2n=1,xy=xk>,k(?)1(mod2),k2≡1(mod11)；(2)<x,y,z|x4=x4=z3=1,x2=y2,xy=x-1,x=y±1,yz=y-1x=1>；<x,y,z|x4=z3=1,x2=y2,xy=x-1,xz=y±1,yz=yx=1>;<x,y,z|x4=z3=1,x2=y2,xy=x-1,xz=y±1x-1,yz=x±1>；<x,y,z|x4=z3=1,x2=y2,xy=x-1,xz=y±1x,yz=x±1>;(3)<x,y|x11=y5n=1,xy=xk>,k(?)1(mod5),k5≡1(mod11)；(4)<x,y,z|x5n=y2=z3=1,[y,z]=1,yx=ys,zx=zt>,s5≡1(mod2), k5≡1(mod3)；(5)<x,y|x11n-1=y11=1,xy=x1+11n-2>,n≥3...