Central Factor Group Research Articles

An Upper Bound for the Index of the Center in Capable Groups with Finite Cyclic Derived Subgroups

A group G is said to be capable if it occurs as the central factor group H/Z(H) for some group H. Motivated by the results of Isaacs [11], in Proc. Amer. Math. Soc. 129(10) (2001), pp. 2853-2859, we show that if G is a capable group with cyclic derived subgroup G′ of odd order, then |G/Z(G)| divides |(G/L)′|2 ϕ(|L|)|L|, in which ϕ is Euler’s function and L is the smallest term of the lower central series of G. Moreover, there is no such capable nonnilpotent group G that holds |G/Z(G)| = |G′|2. In particular, |G/Z(G)| = |G′|2 if and only if G is nilpotent.

Some properties of left 2α-Engel subgroups

In 1904, Schur proved his famous result which says that if the central factor group of a given group is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In the present paper, for a given automorphism [Formula: see text] of the group [Formula: see text], we introduce the concept of left [Formula: see text]-Engel, [Formula: see text], and [Formula: see text]-Engel commutator, [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

Finite groups which are c-absolute central factor group and the union of Ac-centralizers

Let [Formula: see text] be a group and [Formula: see text] be the [Formula: see text]-absolute center of [Formula: see text], that is, the set of all elements of [Formula: see text] fixed by all class preserving automorphisms of [Formula: see text]. In this paper, we classify all finite groups [Formula: see text], whose [Formula: see text]-absolute central factors are isomorphic to the direct product of cyclic groups, [Formula: see text] and [Formula: see text]. Moreover, we consider finite groups which can be written as the union of centralizers of class preserving automorphisms and study the structure of [Formula: see text] for groups, in which the number of distinct centralizers of class preserving automorphisms is equal to 4 or 5.

Some results of left and right α-commutators of groups

In 1904, Schur proved his famous result which says that if the central factor group of a given group [Formula: see text] is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In this paper, we introduce the concept of left and right [Formula: see text]-commutator, [Formula: see text], and [Formula: see text], where [Formula: see text] is an automorphism of the group [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

On the capability of finite p-groups with derived subgroup of order p

A group is said to be capable if it is a central factor group. Let denotes the class of finite p-groups with derived subgroup of order p and central factor group of order p2. In this paper for groups in , we compute the various homological functors, among them the nonabelian tensor square and the Schur multiplier. Furthermore, the epicenters of all these groups are determined to give a complete classification of finite capable p-groups with derived subgroup of order p.

Central Factor Groups and Commutativity

Central Factor Groups and Commutativity

Central Factor Groups and Commutativity

Central Factor Groups and Commutativity

The theorems of Schur and Baer: a survey

This paper gives a short survey of some of the known results generalizing the theorem, credited to I. Schur, that if the central factor group is nite then the derived subgroup is also nite.

Finite groups with a given absolute central factor group

Let G be a group and L(G) be the absolute center of G, that is, the set of all elements of G fixed by all automorphisms of G. In this paper, we classify all finite groups G whose absolute central factors are isomorphic to a cyclic group, \({\mathbb{Z}_p \times \mathbb{Z}_p}\) , D8, Q8, or a non-abelian group of order pq for some distinct primes p and q.

The ranks of central factor and commutator groups

AbstractThe Schur Theorem says that if G is a group whose center Z(G) has finite index n, then the order of the derived group G′ is finite and bounded by a number depending only on n. In this paper we show that if G is a finite group such that G/Z(G) has rank r, then the rank of G′ is r-bounded. We also show that a similar result holds for a large class of infinite groups.

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 the Norm and Wielandt Series in Finite Groups

The norm N(G) of a group G is the intersection of the normalizes of all the subgroups of G. A group is called capable if it is a central factor group. In this paper, we give a necessary and sufficient condition for a capable group to satisfy N(G)=ζ(G), and then some sufficient conditions for a capable group with N(G)=ζ(G) are obtained. Furthermore, we discuss the norm of a nilpotent group with cyclic derived subgroup.

On the nonabelian tensor square and capability of groups of order p 2 q

A group G is said to be capable if it is isomorphic to the central factor group H/Z(H) for some group H. Let G be a nonabelian group of order p2q for distinct primes p and q. In this paper, we compute the nonabelian tensor square of the group G. It is also shown that G is capable if and only if either Z(G) = 1 or p < q and \({G^{\rm ab}=\mathbb{Z}_{p}\times\mathbb{Z}_{p}}\) .

ON THE CAPABILITY OF FINITELY GENERATED NON-TORSION GROUPS OF NILPOTENCY CLASS 2

AbstractA group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

ON FINITE p-GROUPS WITH SUBGROUPS OF BREADTH 1

AbstractWe consider finite p-groups G in which every cyclic subgroup has at most p conjugates. We show that the derived subgroup of such a group has order at most p2. Further, if the stronger condition holds that all subgroups have at most p conjugates then the central factor group has order p4 at most.

The exponents of central factor and commutator groups

The exponents of central factor and commutator groups

Capable 2-Generator 2-Groups of Class Two

A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.

Capability of nilpotent products of cyclic groups

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpo tent groups of class 2. We also establish a necessary condition for the capability of an arbitrary p -group of class k, and some further results.

On the orders of generators of capablep-groups

A group is called capable it if is a central factor group. For each primepand positive integerc, we prove the existence of a capablep-group of classcminimally generated by an element of orderpand an element of orderp1+⌊c−1/p−1⌋. This is best possible.

On capable $p$-groups of nilpotency class two

A group is called capable if it is a central factor group. Let ${\mathcal{P}}$ denote the class of finite $p$-groups of odd order and nilpotency class 2. In this paper we determine the capable 2-generator groups in ${\mathcal{P}}$. Using the explicit knowledge of the nonabelian tensor square of 2-generator groups in ${\mathcal {P}}$, we first determine the epicenter of these groups and then identify those with trivial epicenter, making use of the fact that a group has trivial epicenter if and only if it is capable. A capable group in ${\mathcal{P}}$ has the two generators of highest order in a minimal generating set of equal order. However, this condition is not sufficient for capability in ${\mathcal{P}}$. Furthermore, various homological functors, among them the exterior square, the symmetric square and the Schur multiplier, are determined for the 2-generator groups in ${\mathcal{P}}$.