Finite $$p$$ p -Groups all of Whose Subgroups of Index $$p^3$$
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Finite p-Groups all of Whose Subgroups of Index p3 are Abelian Qinhai Zhang1 · Libo Zhao1 · Miaomiao Li1 · Yiqun Shen1
Received: 1 March 2015 / Revised: 15 March 2015 / Accepted: 17 March 2015 / Published online: 4 April 2015 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2015
Abstract Suppose that G is a finite p-group. If all subgroups of index p t of G are abelian and at least one subgroup of index p t−1 of G is not abelian, then G is called an At -group. We use A0 -group to denote an abelian group. From the definition, we know every finite non-abelian p-group can be regarded as an At -group for some positive integer t. A1 -groups and A2 -groups have been classified. Classifying A3 -groups is an old problem. In this paper, some general properties about At -groups are given. A3 -groups are completely classified up to isomorphism. Moreover, we determine the Frattini subgroup, the derived subgroup and the center of every A3 -group, and give the number of A1 -subgroups and the triple (μ0 , μ1 , μ2 ) of every A3 -group, where μi denotes the number of Ai -subgroups of index p of A3 -groups. Keywords
Finite p-groups · Minimal non-abelian p-groups · At -groups
Mathematics Subject Classification
20D15
1 Introduction Finite p-groups form an important class of finite groups. After the classification of finite simple groups was finally completed, the study of finite p-groups becomes more and more active. Many leading group theorists, for example, Glauberman and Janko have turned their attention to the study of finite p-groups. Quite different to finite simple groups, it is impossible to classify finite p-groups in the classical sense. The
B 1
Qinhai Zhang [email protected]; [email protected] Department of Mathematics, Shanxi Normal University, Linfen 041004, Shanxi, People’s Republic of China
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Q. Zhang et al.
reason is that a finite p-group has “ too many” normal subgroups and consequently there is an extremely large number of non-isomorphic p-groups of a given fixed order. In fact, Higman and Sims [14,15,26] gave a formula for the number f (n, p) of nonisomorphic p-groups of order p n : when n → ∞,
f (n, p) = p n
3 (2/27+O(n −1/3 ))
.
It is easy to see that when n becomes larger, the number of non-isomorphic p-groups of order p n becomes larger in exponent speed. For example, up to now, the known results about the number of 2-groups [10] are as follows:
n
2
22
23
24
25
26
27
28
29
210
f (n, 2)
1
2
5
14
51
267
2328
56092
10494213
49487365422
For p > 2, p-groups of order p 7 are classified by [20]. The result is f (7, 3) = 9310, f (7, 5) = 34297. For p > 5, f (7, p) = 3 p 5 +12 p 4 +44 p 3 +170 p 2 +707 p+2455+(4 p 2 +44 p+291) gcd( p− 1, 3) + ( p 2 + 19 p + 135) gcd( p − 1, 4) + (3 p + 31) gcd( p − 1, 5) + 4 gcd( p − 1, 7) + 5 gcd( p − 1, 8) + gcd( p − 1, 9). Because of the difficulty of the classification of finite p-groups in the classical sense, Janko and Berkovich sponsored and led an research pr
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