Finite Nonabelian P-groups Research Articles

ON NONINNER AUTOMORPHISMS OF SOME FINITE P-GROUPS

Abstract We settle the noninner automorphism conjecture for finite p-groups ( $p> 2$ ) with certain conditions. Also, we give an elementary and short proof of the main result of Ghoraishi [‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc.89(2) (2014) 202–209].

The absolute Frattini automorphisms

Let G be a finite non-abelian p-group, where p is a prime number, and Aut(G) be the group of all automorphisms of $G$. An automorphism alpha of $G$ is called absolute central automorphism if, x^{-1}alpha(x) lies in L(G), where L(G) is the absolute center of G. In addition, alpha is an absolute Frattini automorphism if x^{-1}alpha(x) is in Phi(L(G)), where Phi(L(G)) is the Frattini subgroup of the absolute center of G, and let LF(G) denote the group of all such automorphisms of G. Also, we denote by C_{LF(G)}(Z(G)) and C_{LA(G)}(Z(G)), respectively, the group of all absolute Frattini automorphisms and the group of all absolute central automorphisms of G, fixing elementwise the center Z(G) of G . We give necessary and sufficient conditions on a finite non-abelian p-group G of class two such that C_{LF(G)}(Z(G))=C_{LA(G)}(Z(G)). Moreover, we investigate the conditions under which LF(G) is a torsion-free abelian group.

On the coincidence of automorphism groups of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let γn (G) and Zn (G) respectively denote the nth term of the lower and upper central series of G, and let Ln (G) denote the nth absolute center of G. An automorphism α of G is called an IA n -automorphism if x− 1 α(x) ∈ γn +1(G) for all x ∈ G. The group of all IA n -automorphisms of G is denoted by IA n (G). Let denote the subgroup of IA n (G) which fixes Zn (G) elementwise. In this paper, we give necessary and sufficient conditions for a finite non-abelian p-group G of class (n + 1) such that , where M is a subgroup of G such that γn +1(G) ≤ M ≤ Z(G) ∩ Ln (G), and as a consequence, we obtain the main result of Chahal, Gumber and Kalra [7, Theorem 3.1]. We also give necessary and sufficient conditions for a finite non-abelian p-group G of class (n+1) such that Aut z (G) = IA n (G), and obtain Theorem B of Attar [4] as a particular case.

Some necessary and sufficient conditions on commuting automorphisms of some finite p-groups

Let G be a finite non-abelian p-group of order greater than p 4, p an odd prime, such that is cyclic and Z(H) is elementary abelian, where We prove that the set of all commuting automorphisms of G forms a subgroup of if and only if is abelian. Also, we find the structure of for a finite 2-group G of almost maximal class with cyclic center Z(G), where denotes the set of all central automorphisms of G.

Finite p-groups which are non-inner nilpotent

A group G is called a non-inner nilpotent group, whenever it is nilpotent with respect to a non-inner automorphism. In 2018, all finitely generated abelian non-inner nilpotent groups have been classified. Actually, the authors proved that a finitely generated abelian group G is a non-inner nilpotent group, if G is not isomorphic to cyclic groups Z_p_1p_2...p_t and Z, for a positive integer t and distinct primes p_1, p_2,..., p_t. We conjecture that all finite non-abelian p-groups are non-inner nilpotent and we prove this conjecture for finite $p$-groups of nilpotency class 2 or of co-class 2.

On equality of absolute central and IA-automorphism groups of finite p-groups

Let G be a finite p-group. Let and respectively denote the group of all absolute central and -automorphisms of G. We give necessary and sufficient conditions for a finite non-abelian p-group G such that and as a consequence, we prove that for a finite non-abelian p-group G of order up to p 4, if and only if where L(G) denotes the absolute center of G.

Abelian Covers and Non-Commuting Sets in a Non-Abelian p-Group Which its Central Quotient is Metacyclic

Let G be a group. A set S in G is said to be non-commuting if $$xy \ne yx$$ for any two distinct elements $$x,y \in S$$ . We define w(G) to be the maximum possible cardinality of a non-commuting set in G. In this paper, we determine w(G) for a finite non-abelian p-group G such that G/Z(G) is metacyclic by obtaining an abelian centralizers cover of this group. As a consequence, we show that the set of all commuting automorphisms of a finite non-abelian p-group G such that G/Z(G) is metacyclic, forms a subgroup of $$\mathrm {Aut}(G)$$ .

On Commuting Automorphisms of Finite p-Groups with a Metacyclic Quotient

Let G be a finite non-Abelian p-group, where p is an odd prime, such that G/Z(G) is metacyclic. We prove that all commuting automorphisms of G form a subgroup of Aut(G) if and only if G is of nilpotence class 2.

On the Equality of Certain Subgroups of the Automorphism Groups of Finite p-Groups

Let G be a finite non-Abelian p-group, where p is a prime. An automorphism α of G is called an IA-automorphism if x−1α(x) ∈ G′ for all x ∈ G. An automorphism α of G is called an absolute central automorphism if, for all x ∈ G, x−1α(x) ∈ L(G), where L(G) is the absolute center of G. Let CIA(G)(Z(G)) and CVar(G)(Z(G)) denote, respectively, the group of all IA-automorphisms and the group of all absolute central automorphisms of G fixing the center Z(G) of G elementwise. We give necessary and sufficient conditions on a finite p-group G under which CIA(G)(Z(G)) = CVar(G)(Z(G)).

Reduced fusion systems over p-groups with abelian subgroup of index p: III

AbstractWe finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all 𝔽pG-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [15] using the classification of finite simple groups.

Two impartial games on finite p-groups

This paper is about two impartial combinatorial games, namely achievement and avoidance games. These games can be played on groups by two persons. If G is a finite group, then both players chose elements of G alternately. The player who builds a generating set for G is called the loser in avoidance game whereas the builder of generating set is called the winner in achievement game. Dana and Sieben have been provided concept of nim-numbers by plotting their structure diagrams for both games. Since, the nim-numbers provide a more easier analysis of impartial combinatorial games and provide final outcome of these impartial games. But it is hard to calculate the nim-numbers of an impartial game. In this paper, the nim-numbers for achievement and avoidance games on some non-abelian groups have been calculated. In particular, the simplified structure diagrams and nim-values of these two games for finite non-abelian p-groups, dicyclic type groups, and groups of the form ℂ3 × ℂp has been determined.

Finite nonabelian p-groups of exponent >p with a small number of maximal abelian subgroups of exponent >p

Finite nonabelian p-groups of exponent >p with a small number of maximal abelian subgroups of exponent >p

On noninner automorphisms of 2-generator finite p-groups

ABSTRACTA longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p.

On equality of certain automorphism groups of finite groups

ABSTRACTLet G be a finite group and L(G) denotes the absolute center of G. An automorphism α of G is called an autocentral automorphism if x−1α(x)∈L(G) for each x∈G. An automorphism α of G is called a central automorphism if x−1α(x)∈Z(G) for each x∈G. An automorphism α of G is called an IA-automorphism if for each x∈G. In this paper, we find necessary and sufficient conditions on G such that every autocentral automorphism is inner. Also we characterize finite non-Abelian p-groups for which every central automorphism is IA-automorphism.

On equality of certain automorphism groups

ABSTRACTLet G = H×A be a group, where H is a purely non-Abelian subgroup of G, and A is a non-trivial Abelian factor of G. Then, for n≥2, we show that there exists an isomorphism such that . Also, for a finite non-Abelian p-group G satisfying a certain natural hypothesis, we give some necessary and sufficient conditions for . Furthermore, for a finite non-Abelian p-group G, we study the equality of Autcent(G) with .

Finite p-groups with some isolated subgroups

We say that a subgroup H is isolated in a group G if for each x∈G we have either x∈H or 〈x〉∩H={1}. Here we shall determine certain classes of finite nonabelian p-groups which possess some isolated subgroups (Theorems 1, 2, 4, 6 and 8 to 16).

On Finite p-Groups Whose IA-Automorphisms Are All Class Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism α of G is called a class preserving automorphism if α(x) ∈ x G the conjugacy class of x in G, for all x ∈ G. An automorphism α of G is called an IA-automorphism if x −1α(x) ∈ G′ for each x ∈ G. In this paper, we give necessary and sufficient conditions on finite p-group G of nilpotency class 2 such that every IA-automorphism is class preserving.

Finite p-groups of exponent [formula omitted] all of whose cyclic subgroups of order [formula omitted] are normal

Here we classify finite nonabelian p-groups G of exponent pe, e≥2, all of whose cyclic subgroups of order pe are normal in G. Let G0 be the subgroup of G generated by all elements of order pe. If p>2, then G0=G and G is of class 2 (Theorem 1). However if p=2, then |G:G0|≤2 and in case |G:G0|=2 the structure of G is more complicated (Theorems 3, 4 and 5).

Finite nonabelian p-groups all of whose subgroups are q-self dual

A finite p-group G is q-self dual if every quotient of G is isomorphic to a subgroup of G. Here, we determine finite 2-groups G all of whose subgroups are q-self dual (Theorem 3) and in case p > 2 we get a classification of such groups only under the additional assumptions that Ω1(G) is abelian (Theorem 4).

ON THE CENTER OF THE MODULAR GROUP ALGEBRA OF A FINITE p-GROUP

Let G be a finite nonabelian p-group and F a field of characteristic p and let [Formula: see text] be the subalgebra spanned by class sums [Formula: see text], where C runs over all conjugacy classes of noncentral elements of G. We show that all finite p-groups are subgroups and homomorphic images of p-groups for which [Formula: see text]. We also give the description of abelian-by-cyclic groups for which [Formula: see text] is an algebra with zero multiplication or is nil of index 2.