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 nonnormal subgroups have orders at most p^3

We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. 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 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.

A Characteristic of Dedekind Groups

In this paper, we show that if a finite group G has a fixed-point-free weak power automorphism, then G is a Dedekind group.