The Study Of Properties Of Groups In Which All Non-subnormal Subgroups Forming A Class Of Conjugate

In this paper, we investigate some properties of the groups in which all nonsubnormal subgroups forming a class of conjugation. We denote the number of the conjugate classes of non-subnormal subgroups of G byμ, and have the following results:Theorem 2.1 G is a finite group withμ= 1 if and only if where f(x) = x^β-d_βx^β-1-…-d₂x - d₁ is an irreducible polynomial over the field F_q, which divides x^p - 1.Theorem 2.2 If the finite group G has two conjugate classes of non-subnormal subgroups H= H₁,H₂,…, H_m and K=K_i, K₂,…, K_n, then G has the following properties:(1) There is at least one non-subnormal subgroup being Sylow-subgroup of G ;(2) There is only one conjugate class of non-subnormal subgroups which are maximal in G . If H =H₁, H₂,…, H_m are maximal, then K=K₁, K₂,…, K_n are cyclic p-subgroups. Whatmore, if K_i∈Syl_p(G), and p is the smallest prime factor of |G|, then G is p-nilpotent, and so G is p-normal.(3) G must contain maximal normal subgroup, be soluble and | G | has at most three prime factors. Theorem 3.1 Let G be a group. Ifμ= 1 and the length of the conjugate class is finite, then G is a finite non-nilpotent inner-Abel group.Theorem 3.2 Let G be a CF-group. Then the following conditons are equivalent:(1) G is nilpotent;(2) All subgroups of G are subnormal;(3) All maximal subgroups of G are normal;(4) G satisfies the normalizer condition.Theorem 4.1 There is no exist a locally nilpotent group satisfing one of the following conditions:(1) The non-subnormal subgroups of G are finite;(2) H is a non-subnormal subgroup of G and there is no element of infinite order g∈G satisfing H≠H^g;(3) The set of non-subnormal subgroups satisfies the maximal condition or minimal condition.Theorem 4.2 There is no exist a non-tortion group satisfing one of the following conditions:(1) The non-subnormal subgroups of G are finite;(2) H is a non-subnormal subgroup of G and there is no element of infinite order g∈G satisfing H≠H^g;(3) The set of non-subnormal subgroups satisfies the maximal condition or minimal condition.Theorem 4.3 There is no exist an infinite locally finite group withμ=1.Theorem 4.4 There is no exist a non-commuting inner Abel-group withμ= 1.