Cyclic Normal Subgroup Research Articles

Class-preserving Coleman automorphisms of some classes of finite groups

Abstract The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G G be a finite group with a nilpotent subgroup N N . Suppose that G / N G\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of N N . Then it is proved that every class-preserving Coleman automorphism of G G is an inner automorphism. In addition, if G G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.

Asymptotic performance of metacyclic codes

A finite group is said to be metacyclic if it has a cyclic normal subgroup N such that G∕N is cyclic. A code over a finite field F is called metacyclic if it is a left ideal in the group algebra FG, with G a metacyclic group. Metacyclic codes are generalizations of dihedral codes, and can be constructed as quasi-cyclic codes with an extra automorphism.In this paper, we prove that metacyclic codes form an asymptotically good family of codes. Our proof relies on Artin’s primitive root conjecture for primes in certain arithmetic progressions being true, something that currently can only be guaranteed assuming the Generalized Riemann Hypothesis (GRH).

Two new criteria for solvability of finite groups

We prove the following two new criteria for the solvability of finite groups. Theorem 1: Let G be a finite group of order n containing a subgroup A of prime power index ps. Suppose that A contains a normal cyclic subgroup B satisfying the following condition: A/B is a cyclic group of order 2r for some non-negative integer r. Then G is a solvable group. Theorem 3: Let G be a finite group of order n and suppose that ψ(G)≥16.68ψ(Cn), where ψ(G) denotes the sum of the orders of all elements of G and Cn denotes the cyclic group of order n. Then G is a solvable group.

On the Ultrasolvability of p-Extensions of an Abelian Group by a Cyclic Kernel

The paper contains a solution of A. V. Yakovlev’s problem in the embedding theory for p-extensions of odd order with a cyclic normal subgroup and an Abelian quotient group: for such nonsplit extensions there exists a realization for the quotient group as a Galois group over number fields such that the corresponding embedding problem is ultrasolvable (i.e., this embedding problem is solvable and has only fields as solutions). A solution for embedding problems of p-extensions of odd order with kernel of order p and with a quotient group that is represented by a direct product of its proper subgroups is also given – this is a generalization for p > 2 of an analogous result for p = 2 due to A. Ledet.

On Clifford theory with Galois action

Let Gˆ be a finite group, N a normal subgroup of Gˆ and ϑ∈IrrN. Let F be a subfield of the complex numbers and assume that the Galois orbit of ϑ over F is invariant in Gˆ. We show that there is another triple (Gˆ1,N1,ϑ1) of the same form, such that the character theories of Gˆ over ϑ and of Gˆ1 over ϑ1 are essentially “the same” over the fieldF and such that the following holds: Gˆ1 has a cyclic normal subgroup C contained in N1, such that ϑ1=λN1 for some linear character λ of C, and such that N1/C is isomorphic to the (abelian) Galois group of the field extension F(λ)/F(ϑ1). More precisely, having “the same” character theory means that both triples yield the same element of the Brauer–Clifford group BrCliff(G,F(ϑ)) defined by A. Turull.

Permutation-like matrix groups with a maximal cycle of prime square length

A matrix group is said to be permutation-like if any matrix of the group is similar to a permutation matrix. G. Cigler proved that, if a permutation-like matrix group contains a normal cyclic subgroup which is generated by a maximal cycle and the matrix dimension is a prime, then the group is similar to a permutation matrix group. This paper extends the result to the case where the matrix dimension is a square of a prime.

ON FINITE METACYCLIC p-GROUPS

If a metacyclic group G of exponent pe contains a normal cyclic subgroup A of order pe and, in the case p = 2, has no nonabelian sections of order 8, then there exists in G a cyclic subgroup B such that G = AB and A ∩ B = {1}. Some consequences of the stated result are also proved. For example, if a metacyclic group of exponent pe is noncyclic and has no nonabelian sections of order 8 and A < G is cyclic of order pe, then there is a cyclic B < G such that G = AB and A ∩ B = {1}. Next, it is proved that if G > {1} is a metacyclic p-group, then exp (Φ(G)) < exp (G).

On split metacyclic groups

We consider sufficient conditions for metacyclic groups to split. Specifically, we show that a finite metacyclic group G of odd order is split on its cyclic normal subgroup K if K is such that G/K is cyclic and |K| = exp G.

Finite p-groups which contain a self-centralizing cyclic normal subgroup

For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.

CLASS-PRESERVING AUTOMORPHISMS OF GENERALIZED FREE PRODUCTS AMALGAMATING A CYCLIC NORMAL SUBGROUP

In general, a class-preserving automorphism of generalized free products of nilpotent groups, amalgamating a cyclic normal subgroup of order 8, need not be an inner automorphism. We prove that every class-preserving automorphism of generalized free products of nitely generated nilpotent groups, amalgamating a cyclic normal subgroup of order less than 8, is inner.

G-GRAPHS AND SPECIAL REPRESENTATIONS FOR BINARY DIHEDRAL GROUPS IN GL(2,ℂ)

AbstractGiven a finite subgroup G⊂GL(2,ℂ), it is known that the minimal resolution of singularity ℂ2/G is the moduli space Y=G-Hilb(ℂ2) of G-clusters ⊂ℂ2. The explicit description of Y can be obtained by calculating every possible distinguished basis for as vector spaces. These basis are the so-called G-graphs. In this paper we classify G-graphs for any small binary dihedral subgroup G in GL(2,ℂ), and in the context of the special McKay correspondence we use this classification to give a combinatorial description of special representations of G appearing in Y in terms of its maximal normal cyclic subgroup H ⊴ G.

Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three

A group G is metacyclic if it contains a cyclic normal subgroup K such that G/K is also cyclic. Metacyclic p-groups classified by different authors. A group is called capable if it is a central factor group. The purpose of this study is to compute the exterior center of finite non-abelian metacyclic p-groups, p is an odd prime, for some small order groups using Groups, Algorithms and Programming (GAP) software. We also determine which of these groups are capable.

On nonsymmetric theorems for $(H,G)$-coincidences

Let $X$ be a compact Hausdorff space, $\varphi\colon X\to S^{n}$ a continuous map into the $n$-sphere $S^n$ that induces a nonzero homomorphism $\varphi^{*}\colon H^{n}(S^{n};{\mathbb Z}_{p})\to H^{n}(X;{\mathbb Z}_{p})$, $Y$ a $k$-dimensional CW-complex and $f\colon X\to Y$ a continuous map. Let $G$ a finite group which acts freely on $S^{n}$. Suppose that $H\subset G$ is a normal cyclic subgroup of a prime order. In this paper, we define and we estimate the cohomological dimension of the set $A_{\varphi}(f,H,G)$ of $(H,G)$-coincidence points of $f$ relative to $\varphi$.

The intersection map of subgroups and the classes $${\mathcal {PST}_c}$$ , $${\mathcal {PT}_c}$$ and $${\mathcal {T}_c}$$

A group G is called a \({\mathcal {T}_{c}}\)-group if every cyclic subnormal subgroup of G is normal in G. Similarly, classes \({\mathcal {PT}_{c}}\) and \({\mathcal {PST}_{c}}\) are defined, by requiring cyclic subnormal subgroups to be permutable or S-permutable, respectively. A subgroup H of a group G is called normal (permutable or S-permutable) cyclic sensitive if whenever X is a normal (permutable or S-permutable) cyclic subgroup of H there is a normal (permutable or S-permutable) cyclic subgroup Y of G such that \({X=Y \cap H}\). We analyze the behavior of a collection of cyclic normal, permutable and S-permutable subgroups under the intersection map into a fixed subgroup of a group. In particular, we tie the concept of normal, permutable and S-permutable cyclic sensitivity with that of \({\mathcal {T}_c}\), \({\mathcal {PT}_c}\) and \({\mathcal {PST}_c}\) groups. In the process we provide another way of looking at Dedekind, Iwasawa and nilpotent groups.

TORSION UNITS IN INTEGRAL GROUP RINGS OF CERTAIN METABELIAN GROUPS

AbstractIt is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

A Classification of Metacyclic 2-Groups

A finite p-group G is called metacyclic if it has a cyclic normal subgroup A such that G/A is cyclic. In 1973, King classified metacyclic p-groups; in 1987, Newman and the first author developed a new approach to metacyclic p-groups suggested by the p-group generation algorithm, they found new presentations for these groups. However, their results were only announced in the case of p odd, and no proof was given. The purpose of this paper is to give a new proof of their results in the case of p=2, which is independent of the p-group generation algorithm.

On Semiabelian Groups

We establish some structural results on the semiabelian groups, that is, groups generated by their cyclic normal subgroups. Such groups play a significant role in the theory of supersoluble groups. A description of two‐generated semiabelian groups and torsion‐free semiabelian groups is obtained.

FINITE METACYCLIC GROUPS WITH FAITHFUL IRREDUCIBLE REPRESENTATIONS

A metacyclic group is a group that has a cyclic normal subgroup with cyclic quotient. We characterize finite metacyclic groups that have faithful irreducible representations over a given field.

Ovoids of the Hermitian surface in odd characteristic

We construct a new ovoid of the polar space arising from the Hermitian surface of PGð3; q 2 Þ with qd 5 odd. The automorphism group G of such an ovoid has a normal cyclic subgroup F of order 1 ðq þ 1Þ such that G=F G PGLð2; qÞ. Furthermore, G has three orbits on the ovoid, one of size q þ 1 and two of size 1 qðq � 1Þðq þ 1Þ.

On automorphisms of some $p$-groups

A group G is said to enjoy Hasse principle if every local coboundary of G is a global coboundary. Let G be a non-Abelian finite p-group of order p m , p prime and m > 4 having a normal cyclic subgroup of order p m-2 but having no element of order p m-1 . We prove that G enjoys Hasse principle if p is odd but in the case p = 2, there are fourteen such groups twelve of which enjoy Hasse principle but the remaining two do not satisfy Hasse principle. We also find all the conjugacy preserving outer automorphisms for these two groups.