Finite Nonabelian P-groups Research Articles

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Autc(G) and Autz(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Autc(G) = Autz(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Autc(G) = Autz(G). We also characterize all finite p-groups G of order ≤ p 7 such that Autz(G) = Autz(G) and complete the classification of all finite p-groups of order ≤ p 5 for which there exist non-inner class preserving automorphisms.

Simple fusion systems over p-groups with abelian subgroup of index p: I

For an odd prime p, we look at simple fusion systems over a finite nonabelian p-group S which has an abelian subgroup A of index p. When S has more than one such subgroup, we reduce this to a case already studied by Ruiz and Viruel. When A is the unique abelian subgroup of index p in S and is not essential (equivalently, is not radical) in the fusion system, we give a complete list of all possibilities which can occur. This includes several families of exotic fusion systems, including some which have proper strongly closed subgroups.

Commutativity Pattern of Finite Non-Abelian p-Groups Determine Their Orders

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ G (called noncommuting graph of G) with G as follows: Take G∖Z(G) as the vertices of Γ G , and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that “the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that Γ P ≅ Γ H for some group H, then |P| = |H|.

Verification of an Old Conjecture on Nonabelian 2-generated Groups of Order p<sup>3</sup>

A longstanding conjecture in group theory states: “Every finite non-abelian p-group possesses at least a non-inner automorphism of order”, where is a prime number. Recently, an updated classification of 2-generated p-groups of nilpotency class two has been published. Using this classification, we prove the verification of this conjecture for 2-generated groups of order.

Cohomologically trivial modules over finite groups of prime power order

We show the existence of cohomologically trivial Q-module A, where Q = G / Φ ( G ) , A = Z ( Φ ( G ) ) , G is a finite non-abelian p-group, Φ ( G ) is the Frattini subgroup of G, Z ( Φ ( G ) ) is the center of Φ ( G ) , and Q acts on A by conjugation, i.e., z g Φ ( G ) : = z g = g − 1 z g for all g ∈ G and all z ∈ Z ( Φ ( G ) ) . This means that the Tate cohomology groups H n ( Q , A ) are all trivial for any n ∈ Z . Our main result answers Problem 17.2 of [V.D. Mazurov, E.I. Khukhro (Eds.), The Kourovka Notebook. Unsolved Problems in Group Theory, seventeenth edition, Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 2010] proposed by P. Schmid.

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

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

Finite nonabelian p-groups having exactly one maximal subgroup with a noncyclic center

We prove here that a nonabelian finite p-group G has exactly one maximal subgroup with a noncyclic center if and only if Z(G) is cyclic and G has exactly one normal abelian subgroup of type (p, p).

Powerful p-groups have non-inner automorphisms of order p and some cohomology

In this paper we study the longstanding conjecture of whether there exists a non-inner automorphism of order p for a finite non-abelian p-group. We prove that if G is a finite non-abelian p-group such that G / Z ( G ) is powerful then G has a non-inner automorphism of order p leaving either Φ ( G ) or Ω 1 ( Z ( G ) ) elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd p, by showing that the Tate cohomology H n ( G / N , Z ( N ) ) ≠ 0 for all n ⩾ 0 , where G is a finite p-group, p is odd, G / Z ( G ) is p-central (i.e., elements of order p are central) and N◁ G with G / N non-cyclic.

ON THE AUTOMORPHISM GROUP OF A FINITE P-GROUP WITH CYCLIC FRATTINI SUBGROUP

It is conjectured that if G is a finite non-cyclic p-group of order greater than p², then \ |G| divides | Aut(G)|. In this paper we characterize the finite non-abelian p-groups G with cyclic Frattini subgroup for which |Aut(G)|p = |G|.

Finite p-groups of class 2 have noninner automorphisms of order p

We prove that for any prime number p, every finite non-abelian p-group G of class 2 has a noninner automorphism of order p leaving either the Frattini subgroup Φ ( G ) or Ω 1 ( Z ( G ) ) elementwise fixed.

Group algebras with unit group of class $p$

Let V(F_pG) be the group of normalized units of the group algebra F_pG of a finite nonabelian p-group G over the field F_p of p elements. Our goal is to investigate the power structure of V(F_pG), when it has nilpotency class p. As a consequence, we have proved that if G and H are p-groups with cyclic Frattini subgroups and p>2, then V(F_pG) is isomorphic to V(F_pH) if and only if G and H are isomorphic.

On the minimal number of generators of finite non-abelian p-groups having an abelian automorphism group

(1995). On the minimal number of generators of finite non-abelian p-groups having an abelian automorphism group. Communications in Algebra: Vol. 23, No. 6, pp. 2045-2065.

Some new (s,k,?)-translation transversal designs with non-abelian translation group

For λ >1 and many values of s andk, we give a construction of (s,k,λ)-partitions of finite non-abelian p-groups and of Frobenius groups with non-abelian kernel. These groups are associated with translation transversl designs of the same parameters.

Finite non-Abelian p-groups with complemented non-Abelian normal subgroups

Finite non-Abelian p-groups with complemented non-Abelian normal subgroups

On the Number of Automorphisms of a Finite p-group

In this paper we find a new bound for the function g(h), for which |A(G)|p ≧ pn whenever |G| ≧ pg(h) G a finite p-group. The existence of such a function was first conjectured by W. R. Scott in 1954, who proved that g (2) = 3. In 1956 Ledermann and Neumann proved that in the general case of finite groups g(h) ≦ (h – 1)3.-pn−1 + h[10]. Since then, J. A. Green, J. C. Howarth and K. H. Hyde have reduced this bound considerably. The best (least) bound to date for finite p-groups was obtained by K. H. Hyde [9]. He proved that for h ≧ 5 and g(h) = h + 1 for h ≦ 4. For finite non-abelian p-groups, we improve this bound to: for h ≧ 13, g(h) = 2h – 5 for 5 < h ≦ 8, g(h) = h for h ≦ 5 and for 8 < h ≦ 12 we prove that g(9) = 14, g(10) = 17, g(11) = 20, g(12) = 23.

Finite non-Abelian p-groups with complemented non-Abelian subgroups

Finite non-Abelian p-groups with complemented non-Abelian subgroups