| In [3], Berkovich poses the problem of classifying finite p-groups containing a maximal subgroup which is minimal non-abelian. As a first step towards solving this problem, TianzeLi studies the automorphism groups of minimal non-abelian p-groups in [1]. A finite p-group is called minimal non-abelian if it is not abelian but all of its proper subgroups are abelian. This article based on [1], gives the classification of 2-groups containing a maximal subgroup P, where P = P(2, m, 1) = ＜x, y|x2m= y2 = 1,y-1xy = x1+2m-1＞ with m≥2.Theorem. Let G = ＜P, z＞ be a finite 2-group containing a maximal subgroup P, where P = P(2,m,1) = ＜x,y|x2m= y2 = 1,y-1xy = x1+2m-1＞,m ≥ 2. Let z be in G but not in P, then G has the following classification: case 1: when m ≥ 4 and o(2) = 2: G = G1(0), G2(0), G3(O), G4(0), G5(0), C6(0), G7(0), G8(0). case 2: when m ≥ 4 and o(z) ≠ 2; case 2.1: when z2 = x2k: case 2.1.1: when z2 = x2m-1: G = G4(2m-2). case 2.1.2: when z2 = x2k, where k is odd: G = 67(1). case 2.2: when z2 = x2m-1y: G = G9(2m-2),G10(2m-2),G11(2m-2). case 3: when m = 3 and o(2) = 2: G = G1(0),G2(0),G3(0),G5(0),G6(0),G8(0). case 4: when m = 3 and z2 = x2k: case 4.1: when z2 = x4: G = G3(2); case 4.2: when z2 = x2: G = G8(1). case 5: when m = 2 and o(2) = 2: G = G1(0),G5(0),G17(0).case 6: when m = 2 and z2=x: G=G19(1),where k is a positive integer smaller than 2m-1... |
PDF Full Text Request