Cyclic complementary extensions and skew-morphism

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Torsion-free groups having finite automorphism groups. I

In group theory, the cyclic group is a fundamental and seriously studied and understood classes, and is a corner stone in the study of algebraic structures. This paper studies several generalized characteristics of cyclic group with the aim of extending fundamental results and their implications within a broader context and applications to chemistry. We looked at classical properties of cyclic groups, their generation, structure and subgroup behavior. We also explore generalizations such as the decomposition of finite abelian into cyclic subgroup and their behavior under direct product constructions, and the role of cyclicity in automorphism groups and homomorphic images. Using proof-based analysis, we show that these generalized properties reveal deeper structural insights and enable a quick understanding of algebraic systems. Applications to chemistry are also discussed to highlight the practical relevance of cyclic group theory. The paper concludes the directions to future research on cyclic groups in some complex algebraic systems.

In [Tărnăuceanu, M. (2015). Finite groups with a certain number of cyclic subgroups. Amer. Math. Monthly. 122:275–276], Tărnăuceanu described the finite groups G having exactly cyclic subgroups. In [Belshoff, R., Dillstrom, J., Reid, L. Finite groups with a prescribed number of cyclic subgroups. To appear in Communications in Algebra], the authors used elementary methods to completely characterize those finite groups G having exactly cyclic subgroups for Δ = 2, 3, 4 and 5. In this paper, we prove that for any Δ > 0 if G has exactly cyclic subgroups, then and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly cyclic subgroups for .

The main result of this paper is that the outer automorphism group of a free product of finite groups and cyclic groups is semistable at infinity (provided it is one ended) or semistable at each end. In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. We also prove that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism groups of the free product of four finite groups or the free product of a single finite group with a free group of rank two are virtual duality groups of dimension two, in contrast with the above example. Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krstić and Vogtmann.

If G is a group we will write Aut G for the group of all automorphisms of G and Inn G for the normal subgroup of all inner automorphisms of G. Many authors have studied the relationship between the structure of G and that of Aut G, in particular when the latter is finite. This paper is a further contribution to this study. The first results on groups whose automorphism groups are finite were published by Baer in a paper [2] in which he proved that a torsion group has finite automorphism group only if it is finite. Baer also proved that a group with only a finite number of endomorphisms is finite. In 1962 Alperin [1] characterized finitely generated groups with finitely many automorphisms as finite central extensions of cyclic groups. Nagrebeckil [9] discovered in 1972 the important result that in any group with finitely many automorphisms the elements of finite order form a finite subgroup. This of course generalizes Baer's original result. Robinson El0] has given another proof of Nagrebeckfi's Theorem as well as obtaining information on the primes dividing the order of the maximal torsion subgroup. He also characterized the center of a group whose automorphism group is finite and gave a general method for constructing examples. On the other hand there seems to be little hope of obtaining a useful classification of groups whose automorphism groups are finite, even in the abelian case. Indeed, it has been shown be several authors that torsion-free abelian groups with only one non-trivial automorphism the involution x F--~x~ are relatively common (de Groot [5], Fuchs [4], Corner [3]). However, Hallett and Hirsch have adopted a different approach, asking which finite groups can occur as the automorphism groups of torsion-free abelian groups. They have established the following definitive result [-7, 8]:

In earlier work, the authors described the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group. As might be expected, the even order case is considerably more complicated and we have found it necessary to divide it into two parts. This part deals with the situation where we have a finite group $G$ with a cyclic permutable subgroup $A$ satisfying the additional hypothesis that $X$ is permutable in $A_2X$ for all cyclic subgroups $X$ of $G$ (where $A_2$ is the $2$-component of $A$).

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

The factorizations of the finite exceptional groups of lie type

Let G be a finite group. The inclusion graph of cyclic subgroups of G, Ic(G), is the (undirected) graph with vertices of all cyclic subgroups of G, and two distinct cyclic subgroups ⟨a⟩ and ⟨b⟩, are adjacent if and only if ⟨a⟩ ⊂ ⟨b⟩ or ⟨b⟩ ⊂ ⟨a⟩. In this paper, we classify all finite abelian groups, whose inclusion graph is planar. Also, we study planarity of this graph for finite group G, where |π(Z(G))| ≥ 2.

Abelian groups which can occur as the automorphism groups of a finite group have been investigated by many experts. In 1975, Jonah and Konvisser constructed a group of order p to 8 whose automorphism group is Abelian of order p to 16. In 1983, MacHale showed that the Abelian p-groups which can occur as the automorphism groups of a finite group must be of order greater than p to 1. In this paper, we fix our attention on determining all Abelian p-groups of minimal order which can occur as the automorphism group of a finite group and all groups whose automorphism group is the Abelian p-group of minimal order. We obtain many results about Abelian group which can not occur as the automorphism group of a finite group. Especially, we show that for the group possessing an Abelian automorphism group if it possesses a cyclic commutator subgroup then it must be cyclic itself. By these results we conclude the following: (1) if a non-cyclic group of order p to 7 possesses an Abelian automorphism group, then its automorphism group must be of order p to 12; (2) there is no group whose automorphism group is an Abelian group of order less than or equal to p to 11, where p is not equal to 2. Using the two results and some of Morigi's results we solve some problems given by MacHale in 1983. Our techniques of proof might apply to all Abelian p-groups.

The authors study and classify finite groups with their automorphism groups having orders 4pq2, where p and q are primes such that 2 < p < q. In 2013 the first and the third authors classified the nilpotent groups of this kind; here the authors give the classification of those finite non-nilpotent groups with their automorphism groups having orders 4pq2.

In this study, the t-intuitionistic fuzzy normalizer and centralizer of t intuitionistic fuzzy subgroup are proposed. The t-intuitionistic fuzzy centralizer is normal subgroup of t-intuitionistic fuzzy normalizer and investigate various algebraic properties of this phenomena. We also introduce the concept of t-intuitionistic fuzzy Abelian and cyclic subgroups and prove that every t-intuitionistic fuzzy subgroup of Abelian (cyclic) group is t-intuitionistic fuzzy Abelian (cyclic) subgroup. We show that the image and pre-image of t-intuitionistic fuzzy Abelian (cyclic) subgroup are t-intuitionistic fuzzy Abelian (cyclic) subgroup under group homomorphism.

The power graph of a finite group

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems. The fusion system of a finite group G at a prime p encodes the structure of a Sylow-p-subgroup S together with extra information on G-conjugacy within S in category theoretic terms. From work of Alperin and Broue [1] it emerged that fusion systems of finite groups are particular cases of fusion systems of p-blocks of finite groups. In the early 1990’s, Puig introduced the notion of an abstract fusion system on a finite p-group S as a category whose objects are the subgroups of S and whose morphism sets satisfy a list of properties modelled around what one observes in the case of fusion systems of finite groups and blocks. There are ‘exotic’ fusion systems which do not arise as fusion system of a block. Benson [2] suggested that nonetheless any fusion system should give rise to a p-complete topological space which should coincide with the p-completion of the classifying space BG in case the fusion system does arise as fusion system of a finite group G. Broto, Levi and Oliver laid in [4] the homotopy theoretic foundations of such spaces called p-local finite groups and gave in particular a cohomological criterion for the existence and uniqueness of a p-local finite group associated with a given fusion system. While the existence and uniqueness of p-local finite groups for arbitrary fusion systems is still an open problem, there has been in recent years a steadily growing body of work by many authors trying to add to the understanding of fusion systems by extending classical concepts and results on the p-local structure of finite groups (some of which were relevant for the classification of finite simple groups) to all fusion systems. This is also the underlying philosophy of the present note. The following two theorems are Flavell’s Theorem A and Theorem B in [6], extended to arbitrary fusion systems. Our general terminology on fusion systems follows [7]; in particular, by a fusion system we always mean a saturated fusion system. Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α an automorphism of S acting freely on S−{1}, which stabilises F and whose order is a prime number r which does not divide the orders of the automorphism groups AutF(R) for all F-centric radical subgroups R of S. Then F = NF(S). The hypothesis that α stablises F means that for any two subgroups Q, R of S and any morphism φ : Q → R in F the morphism α ◦ φ ◦ α|α(Q) : α(Q) → α(R) is again a morphism in F . Moreover, a subgroup Q of S is called F -centric if CS(Q ) = Z(Q) for all subgroups Q of S such that Q ∼= Q in F ; a subgroup Q of S is called F -radical if Op(AutF (Q)) = AutQ(Q), the group of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined by the automorphism groups in F of F -centric radical subgroups of S.