Classification Of Some Finite Groups

In this paper, we will give the classification of some finite groups, there are two sections in our reslut. For detail, as the following:In Chapter One, we introduce the backgrounds and the present investigations of this paper.In Chapter Two, we classfy the finite non-PE-groups whose every maximal subgroups of even order are PE-groups.Let G be a non-PE-group of even order, but every maximal subgroup of even order are PE-group. Then G is solvable and |Ï€(G)|≤3 and one of the following statements is true:(a) G is a minimal non-PE-group;(b) G = T×M, |T| = 2 and M is a minimal non-PE-group of odd order;(c) G = TM, |T| = 2, M = PQ is a minimal non-PE-group and(c-1) P is a cyclic p-group with [T,P] = 1;(c-2) Q is normal elementary abelian with CQ(T) = 1, where P and Q is a Sylow p-subgroup and Sylow q-subgroup respectively, p and q are distinct odd primes.In Chapter Three, we denote byδ(G) the number of conjugate classes of non-cyclic subgroup of G. we classify the finite group G satisfyδ(G) = 3.Let G be a finite group, thenδ(G) = 3 (?) one of the following statements holds(1) G is an abelian p-group of (p³, p)-type.(2) G≌p³ = b^p = 1, b^-1ab = a^1+P2>.(3) G≌8 = 1, b² = a⁴, b^-1ab=a^-1>.(4) G≌(a, b: a⁴ = b² = 1, b^-1ab = a^-1).(5) G≌Z_p²Ã—Z_q².(6) G≌Q₈×Z_q²(q≠2).(7) G≌p³= b^qm=1, b^-1ab = a^t, t≠1(modp), t^q≡1(modp³)>. (8) G≌p = b^qm = c^r = 1, b^-1ab = a^t, b^-1cb = c^s, [a,c] = 1, t(?) 1(modp), t^q≡1(modp), s (?) 1(modp), s^q = 1(modp)?.(9) G≌H×Z_r², where H = p = b^qm = 1, b^-1ab = a^t = 1, t (?) 1(modp), t^q≡1 (modp²)>).(10) G≌p = b^p = c^q = 1, [a, b] = 1, [b, c] = 1, c^-1ac = a^t, t (?) 1(modp), t^q≡1(modp)>, where q | P-1.(11) G≌p = b^q = c^r = 1, b^-1ab= a^t, c^-1ac = c^s, [b,c] = 1, t (?) 1(modp), s (?) 1(modp)>.(12) G≌[Z_p²]Z_q², where q|p-1.(13) G≌[Q₈]Z₉.(14) G≌P=.b^qn = 1, b^-1ab= a^t,t^q2(?)1(modp), t^q3≡1(modp)>.