Finite nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian

We give here a complete classification of the title groups (Theorem A).     

Representations of elementary abelian p-groups and finite subgroups of fields

Suppose $\mathbb{F}$ is a field of prime characteristic p and E is a finite subgroup of the additive group $(\mathbb{F},+)$. Then E is an elementary abelian p-group. We consider two such subgroups, say E and $E^{\prime }$, to be equivalent if there is an $\alpha \in {\mathbb{F}}^{\times }:=\mathbb{F}?\{0\}$ such that $E=\alpha E^{\prime }$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.     

Nonnormal and minimal nonabelian subgroups of a finite group

Let G be a finite p-group. If p = 2, then a nonabelian group G = Ω₁(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Ω₁(G) has no subgroup isomorphic to S_p²{\Sigma _{{p^2}}}, a Sylow p-subgroup of the symmetric group of degree p ², then it is generated by nonabelian subgroups of order p ³ and exponent p. If p > 2 and the irregular p-group G has < p nonabelian subgroups of order p ^p and exponent p, then G is of maximal class and order p ^p+1. We also study in some detail the p-groups, containing exactly p nonabelian subgroups of order p ^p and exponent p. In conclusion, we prove three new counting theorems on the number of subgroups of maximal class of certain type in a p-group. In particular, we prove that if p > 2, and G is a p-group of order > p ^p+1, then the number of subgroups ≅ ΣS_p²{\Sigma _{{p^2}}} in G is a multiple of p.     

The lower bound of the number of nonabelian subgroups of finite p-groups

In this paper, we give the lower bound of the number of nonabelian subgroups of possible index of finite p-groups, and classify the finite p-groups such that the number of nonabelian subgroups of possible order are exactly the lower bound.     

Finite p-groups whose nonnormal subgroups are metacyclic

For an odd prime p, we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic, and based on the criterion, the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism. This solves a problem proposed by Berkovich.     

Finite p-groups all of whose non-abelian proper subgroups are metacyclic

In this paper, we classify the finite p-groups all of whose non-abelian proper subgroups are metacyclic and answer a question posed by Berkovich. Received: 22 June 2005     

Finite p-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A)/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

Finite non-elementary abelian p-groups whose number of subgroups is maximal

Assume G is a direct product of M _p (1, 1, 1) and an elementary abelian p-group, where M _p (1, 1, 1) = 〈a, b | a ^p = b ^p = c ^p =1, [a,b]=c,[c,a] = [c,b]=1〉. When p is odd, we prove that G is the group whose number of subgroups is maximal except for elementary abelian p-groups. Moreover, the counting formula for the groups is given.     

The enumeration of subgroups in finite p-groups

We generalize the familiar principle of enumeration due to Hall and establish a new principle for the enumeration of subgroups of any p-group G of order p^m, based on the following grouptheoretic relation found by the author: $$\sum\nolimits_{\lambda = 0}^m {\left( { - 1} \right)^\lambda p^{\left( {\begin{array}{*{20}c} \lambda \\ 2 \\ \end{array} } \right)} \mathcal{E}_\lambda \left( G \right)} = 0,$$ where ?_λ (G) is the number of elementary Abelian subgroups of order p^λ in G.     

On maximal cyclic subgroups in finite p-groups

In this note we consider finite noncyclic p-groups G all of whose maximal cyclic subgroups X satisfy one of the following two properties. (a) If each subgroup H of G containing X properly is nonabelian, then p = 2 and G is generalized quaternion. (b) If X is contained in exactly one maximal subgroup of G, then G is metacyclic. This solves the problems Nr.1541 and Nr. 1594 from [1].     

特定阶的子群都同构且交换的有限p-群

研究某些子群同构的有限p-群是很有趣的.例如,Hermann和Mann都曾研究过极大子群都同构的有限p-群,但这类群的结构非常复杂,到现在人们都没能给出其分类.研究了特定阶的子群都同构且交换的有限p-群,并给出其分类.     

Isolated subgroups of finite abelian groups

Czechoslovak Mathematical Journal - We say that a subgroup H is isolated in a group G if for every x ∈ G we have either x ∈ H or 〈x〉 ∩ H = 1. We describe the set of...     

Infinite nonabelian groups with minimal condition for non-normal subgroups

Infinite nonabelian groups which do not possess a descending chain of non-normal subgroups are examined. It is established that under certain additional conditions, in particular the condition of the existence of a normal system with finite factors, the class of all such groups consists only of infinite extremal nonabelian groups and infinite hamiltonian groups, see [6].Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 11–18, July, 1969.     

Conjugacy classes of non-normal subgroups of finite p-groups

Let G be a finite p-group, and let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. It is known that either ν(G) ≤ 1 or ν(G) ≥ p. We determine all p-groups G with ν(G) ≤ p + 1.