Semicomplete Finite p-Groups of Class 2

2016 ◽  
Vol 23 (04) ◽  
pp. 651-656
Author(s):  
M. Shabani Attar
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Inner Automorphism

Let G be a group and G' be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x-1 α(x) ∈ G' for each x ∈ G. The set of all IA-automorphisms of G is denoted by IA (G). A group G is called semicomplete if and only if IA (G)= Inn (G), where Inn (G) is the inner automorphism group of G. In this paper we completely characterize semicomplete finite p-groups of class 2; we also classify all semicomplete finite p-groups of order pn (n ≤ 5), where p is an odd prime. This completes our work in 2011.

Semicomplete Finite p-Groups

2011 ◽  
Vol 18 (spec01) ◽  
pp. 937-944 ◽  
Author(s):  
M. Shabani Attar
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Necessary Conditions ◽  
Abelian Groups ◽  
Inner Automorphism

Let G be a group and G' be its commutator subgroup. An automorphism α of G is called an IA-automorphism if x-1α (x) ∈ G' for each x ∈ G. The set of all IA-automorphisms of G is denoted by IA (G). A group G is called semicomplete if and only if IA (G)= Inn (G), where Inn (G) is the inner automorphism group of G. In this paper we characterize semicomplete finite p-groups of class 2, give some necessary conditions for finite p-groups to be semicomplete, and characterize semicomplete non-abelian groups of orders p4 and p5.

scholarly journals Transitive action on finite points of a full shift and a finitary Ryan’s theorem

2017 ◽  
Vol 39 (06) ◽  
pp. 1637-1667 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Turing Machines ◽  
Finite Support ◽  
Transitive Action ◽  
Group Invariant ◽  
Finite Set ◽  
Transitivity Property ◽  
Full Shift ◽  
Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s  $V$ .

scholarly journals On the capability of groups

1998 ◽  
Vol 41 (3) ◽  
pp. 487-495 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Automorphism Group ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Integral Homology ◽  
The Third ◽  
Inner Automorphism

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.

scholarly journals Foundations of topological racks and quandles

2016 ◽  
Vol 25 (03) ◽  
pp. 1640002 ◽  
Author(s):  
Mohamed Elhamdadi ◽  
El-Kaïoum M. Moutuou
Keyword(s):  
Automorphism Group ◽  
Cohomology Theory ◽  
Central Extensions ◽  
Continuous Cohomology ◽  
Inner Automorphism

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step toward a general continuous cohomology theory for topological racks and quandles.

scholarly journals On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

Blind Signature Scheme over Inner Automorphism Group

2012 ◽  
Vol 6 (19) ◽  
pp. 538-545
Author(s):  
Ping Pan ◽  
Licheng Wang ◽  
Chenqian Xu ◽  
Yixian Yang
Keyword(s):  
Automorphism Group ◽  
Blind Signature ◽  
Signature Scheme ◽  
Inner Automorphism

scholarly journals TOWARD A CLASSIFICATION OF FINITE QUANDLES

2008 ◽  
Vol 17 (04) ◽  
pp. 511-520 ◽  
Author(s):  
G. EHRMAN ◽  
A. GURPINAR ◽  
M. THIBAULT ◽  
D. N. YETTER
Keyword(s):  
Automorphism Group ◽  
Structure Theorem ◽  
Decomposition Theorem ◽  
New Approach ◽  
Inner Automorphism ◽  
Fourth Author ◽  
Student Team

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

scholarly journals ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP

2013 ◽  
Vol 23 (05) ◽  
pp. 1063-1077 ◽  
Author(s):  
VIVEK K. JAIN ◽  
PRADEEP K. RAI ◽  
MANOJ K. YADAV
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Frattini Subgroup ◽  
First Time

We construct, for the first time, various types of specific non-special finite p-groups having abelian automorphism group. More specifically, we construct groups G with abelian automorphism group such that γ2(G) < Z(G) < Φ(G), where γ2(G), Z(G) and Φ(G) denote the commutator subgroup, the center and the Frattini subgroup of G respectively. For a finite p-group G with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) Z(G) = Φ(G) is elementary abelian; (ii) γ2(G) = Φ(G) is elementary abelian, where p is an odd prime. We construct examples to show the existence of groups G with elementary abelian automorphism group for which exactly one of the above two conditions holds true.

Nielsen's commutator test for two-generator groups

1993 ◽  
Vol 114 (2) ◽  
pp. 295-301 ◽  
Author(s):  
Narain Gupta ◽  
Vladimir Shpilrain
Keyword(s):  
Free Group ◽  
Final Section ◽  
Commutator Subgroup ◽  
Metabelian Group ◽  
Free Metabelian Group ◽  
Sufficiency Condition ◽  
Inner Automorphism ◽  
Tame Automorphism ◽  
Polynilpotent Group ◽  
Inner Automorphisms

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

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