Finite p-Groups in Which the Number of Subgroups of Possible Order Is Less Than or Equal to p^3

In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.

Finite p-Groups all of Whose Subgroups of Index p^(3) are Abelian

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 A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups.

Finite 2-groups whose nonnormal subgroups have orders at most 2^3

In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.

Finite p-Groups Whose Subgroups of Given Order are Isomorphic and Minimal Non-abelian

Finite p-groups whose subgroups of given order are isomorphic and minimal non-abelian are classified. In addition, two results on a chain condition of At-groups are improved.

Some unsolvable conjectures in finite p-groups

We survey some unsolvable conjectures in finite p-groups and their research progress.

Finite p-Groups all of Whose Maximal Subgroups Either are Metacyclic or Have a Derived Subgroup of Order ≤ p

The groups as mentioned in the title are classified up to isomorphism. This is an answer to a question proposed by Berkovich and Janko.

Finite 2-groups whose length of chain of aonnormal subgroups is at most 2

Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept： we say that H1 〈 H2〈.. 〈 Hk is a chain of nonnormal subgroups of G if each Hi G and if ｜Hi ： Hi-1｜ = p for i = 2, 3,..., k. k is called the length of the chain, chn（G） denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn（G） ≤ 2 are completely classified up to isomorphism.