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



Abdulqader Mohammed Abdullah Bin Basri, Nor Haniza Sarmin , Nor Muhainiah Mohd Ali and James R. Beuerle
 
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ABSTRACT

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.

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  How to cite this article:

Abdulqader Mohammed Abdullah Bin Basri, Nor Haniza Sarmin , Nor Muhainiah Mohd Ali and James R. Beuerle, 2012. Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three. Journal of Applied Sciences, 12: 1608-1612.

DOI: 10.3923/jas.2012.1608.1612

URL: https://scialert.net/abstract/?doi=jas.2012.1608.1612
 
Received: March 01, 2012; Accepted: July 10, 2012; Published: August 02, 2012



INTRODUCTION

A group G is metacyclic if there is a cyclic normal subgroup K whose quotient group G/K is also cyclic. The presentation of finite metacyclic groups contains two generators and three defining relations. Much attention have been given to some specific types of metacyclic groups. Metacyclic groups with cyclic commutator quotient discussed by Zassenhaus and Hall. Metacyclic p-groups of odd order have been classified by Beyl, King, Liedahl, Newman and Xu, Rdei and Lindenberg. The result that every finite metacyclic group can be decomposed naturally as a semidirect product of two Hall subgroups made by Sim (1994) was an important progress (Hempel, 2000).

Beuerle (2005) classified all finite non-abelian metacyclic p-groups. The main objective 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 by using Groups, Algorithms and Programming (GAP) software, based on Beuerle’s classification.

Before we introduce the exterior center for a group, we need to introduce the non-abelian tensor square as follows:

Definition 1 (Brown et al., 1987): For a group G, the non-abelian tensor square, G⊗G, is generated by the symbols gqh where g, hεG subject to the relations:

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

for all g, g’, h, h’εG, where hg = hgh-1 denotes the conjugate of g by h.

The exterior square is a factor group of the tensor square, defined as follows:

Definition 2 (Brown et al., 1987): For any group G the exterior square of G is defined as G∧G (G⊗G)'L(G), where Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three is a central subgroup of G⊗G.

The tensor center of a group was defined by Ellis (1995) as:

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

Similarly, the exterior center of a group is defined as:

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

Here, 1q and 1 denote the identities in G⊗G and G∧G, respectively. It can be easily shown that Zq (G) and Z (G) are characteristic and central subgroups of G with Zq (G)⊆Z (G).

Baer (1938) initiated a systemic investigation of the question in which conditions a group G must fulfill in order to be the group of inner automorphisms of some group H, i.e., G≅H⁄Z(H). Baer was the first to study the notion of the capability of a group. Hall and Senior (1964) introduced the term by defining a capable group as equal to its central factor group. The capability for some groups has been studied by many authors including Baer (1938) who characterized finitely generated abelian groups which are capable in the following theorem:

Theorem 1 (Baer, 1938): Let A be a finitely generated abelian group written as Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three, such that each integer ni+1 is divisible by ni and Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three the infinite cyclic group. Then A is capable if and only if k≥2 and nk-1 = nk.

A necessary and sufficient condition for a group to be a capable has been established by Beyl et al. (1979) in terms of the epicenter, defined as follows:

Definition 3 (Beyl et al., 1979): The epicenter, Z* (G) of a group G is defined as:

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

The epicenter can be easily seen as a characteristic subgroup of G contained in its center. The following criterion now characterizes capable groups.

Theorem 2 (Beyl et al., 1979): A group G is capable if and only if Z* (G) = 1.

Beyl et al. (1979) used this characterization to describe the capable finite metacyclic groups and determine the extra special p-groups which are capable.

By Beuerle (2005), a metacyclic p-group is usually given by a presentation of the form:

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

where, m, n≥0, r>0, k≤pm, pm|k(r-1) and Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three.

Metacyclic p-groups are divided by Beuerle (2005) into two main classes as follows:

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

where, r = pα-δ+1 or r = pα-δ-1. We say that the group is of positive or negative type if r = pα-δ+1 or r = pα-δ-1, respectively.

For short, we write G+ and G- for Gp (α, β, ε, δ, +) and Gp (α, β, ε, δ, -), respectively.

The following definition gives a standardized parametric presentation of metacyclic p-groups.

Definition 4 (Beuerle, 2005): Let Gbe a finite metacyclic p-group. Then there exist integers α, β, δ, ε with α, β>0 and δ, ε nonnegative, where δ≤min {α-1, β} and δ+ε≤α such that for an odd prime p, Gp (α, β, ε, δ, +). If p = 2, then in addition α-δ>1 and G2 (α, β, ε, δ, +) or Gp (α, β, ε, δ, -), where in the second case ε≤1. Moreover, if G is of positive type, then δ>0 for all p.

Some of the basic properties and various subgroups are given in the next proposition:

Proposition 1 (Beuerle, 2005): Let G be a group of type Gp (α, β, ε, δ, ±) and k≥1. Then we have the following results for the order of G, the center of G, the order of the center of G, the derived subgroups of G, the order of the derived subgroups of G and the exponent of G:

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

Moreover, if β>max {1,δ} then Z(G_) is cyclic if and only if ε = 1.

Though the criterion for capability stated in theorem 2 is easily formulated, the implementation is another matter. As in all cases before, this still requires the cumbersome process of evaluating the factor groups. The connection between the epicenter and the exterior center given by Ellis (1998) provides a much more efficient procedure to determine capability once the non-abelian tensor square is known. The desired external characterization of the epicenter is obtained as follows:

Theorem 3 (Ellis, 1998): For any group G, the epicenter coincides with the exterior center, i.e., Z* (G) = Z (G).

Several authors used the tensor squares method to determine the capability. Beuerle and Kappe (2000) characterized infinite metacyclic groups which are capable.

By using the criterion stated in Theorem 2 and the relationship between the epicenter and exterior center stated in theorem 3, we determine which of them are capable.

Groups, Algorithms and Programming (GAP) software is used as a tool to verify the results found in determining the capability of the groups studied. GAP is a system for computational discrete algebra, with emphasis on computational group theory. GAP provides a programming language, a library of functions that implement algebraic algorithms written in the GAP language as well as libraries of algebraic objects such as for all non-isomorphic groups up to order 2000 (Rainbolt and Gallian, 2010).

THE CLASSIFICATION OF METACYCLIC p-GROUPS OF NILPOTENCY CLASS AT LEAST THREE

In this section, the classification of all finite non-abelian metacyclic p-groups of nilpotency class at least 3 where p is an odd prime, done by Beuerle is stated.

Theorem 4 (Beuerle, 2005): Let p be an odd prime and G a metacyclic p-group of nilpotency class at least three. Then G is isomorphic to exactly one group in the following list:

Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three
(1)

where, α, β, Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three, δ-1≤α<2δ, δ≤β.

Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three
(2)

where α, β, Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three, δ-1≤α<2δ, δ≤β+ε.

If p = 3 and α = β = δ = 1 satisfying the conditions of type 1, then the group G≅〈a, b; a3 = b3 = 1, [b,a] = a〉 is not metacyclic.

For type 2, if p = 3, α = 5 and β = δ = ε = 3 satisfying the conditions of the type, then the group G≅〈a, b; a243 = 1, b27 = a9, [b, a] = a9〉 is not metacyclic. Therefore, not all groups of type 1 and 2 are metacyclic groups.

Now, we just need to consider the conditions of metacyclic p-groups stated in definition 4 in order to come up with restrictions in the parameters so that nonmetacyclic groups of type 1 and 2 reduced. So, using definition 4, theorem 4 can be straightforwardly rewritten as follows:

Theorem 5: Let p be an odd prime and G a metacyclic p-group of nilpotency class at least three. Then G is isomorphic to exactly one group in the following list:

Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three
(3)

where, α, β, Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three, δ≤α<2δ, δ≤β, δ≤β, δ≤min {α-1, β}

Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three
(4)

where, α, β, Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three, δ+ε≤α<2δ, δ≤β, α<β+ε, δ≤min {α-1, β}.

EXTERIOR CENTER COMPUTATIONS

Here, we use GAP to develop appropriate coding for computing the exterior center for some small order groups of type 3 and 4. We provide GAP programmes to generate general codes and examples of groups of type 3 and 4.

Type 3: First we use GAP programme to generate all finite non-abelian metacyclic p-groups of type 3 and construct some examples.

Generating type 3: GAP coding to generate all finite non-abelian metacyclic p-groups of type 3 is developed as follows:

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

We define the group with four parameters which are p, α, β and δ. The GAP code below is used to create the presentation of the group.

F: = Free group (2)
a: = F.1;b:=F.2
G: = Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three

Constructing examples of type 3: Now, we define the parameters and put them in GAP. For example, we define p = α = 3, β = 2, 3, 4 and δ = 2. First we need to read the file and call the function.

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

Using the coding as before, we got the following results.

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

Summary of results for type 3: For short, we write Ex (G) for the exterior center of G. A summary of GAP results for type 3 is given in Table 1.

Type 4: With similar objective as for groups of type (3), we produce GAP algorithms for groups of type (4).

Generating type 4: We develop the GAP coding to generate all finite non-abelian metacyclic p-groups of type 4.

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

We define the group with five parameters which are p, α, β, ε and δ. The GAP code below is used to create the presentation of the group.

F: = Free group (2);

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

Constructing examples of type 4: Now, we define the parameters and put them in GAP. For example, we define p = α = 3, β = 3, 4, 5, 6, ε = 1 and δ = 2. First we need to read the file and call the function.

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

Table 1: GAP results for type 3
Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three

Table 2: GAP results for type 4
Image for - Computing the Exterior Center of Metacyclic p-groups of Nilpotency Class at Least Three

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

Summary of results for type 4: A summary of GAP results for type (4) is given in Table 2.

CAPABILITY DETERMINATION

Table 1 shows that the groups of type 3 have trivial exterior center with α = β. Table 2 shows that all groups of type 4 have nontrivial exterior center. Therefore, by Theorem 2 and Theorem 3, we conclude from Table 1 that group 2 and group 4 of type 3 are capable and the rest are not capable. Similarly, from Table 2 we conclude that all groups of type 4 are not capable.

CONCLUSION

In this study we have computed the exterior center of finite non-abelian metacyclic p-groups, p is an odd prime, for some small order groups and determined their capability. We showed that the exterior center of groups of type 3 is trivial with α = β and the exterior center of all groups of type 4 is nontrivial. Also, we showed that the groups of type 3 are capable with α = β whereas all the groups of type 4 are not capable. It is worth to mention that the order of all groups computed satisfies pα+β as stated in proposition 1.

ACKNOWLEDGMENT

The authors would like to acknowledge Universiti Teknologi Malaysia for the partial funding of this research through the Research University Grant (RUG) vote No. 05J46.

REFERENCES

1:  Sim, H.S., 1994. Metacyclic groups of odd order. Proc. London Math. Soc., 69: 47-71.

2:  Hempel, C.E., 2000. Metacyclic groups. Bull. Austr. Math. Soc., 61: 523-524.

3:  Beuerle, J. and L.C. Kappe, 2000. Infinite metacyclic groups and their nonabelian tensor squares. Proc. Edinburgh Math. Soc., 43: 651-662.
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4:  Brown, R., D.L. Johnson and E.F. Robertson, 1987. Some computations of nonabelian tensor products of groups. J. Algebra, 111: 177-202.

5:  Ellis, G., 1995. Tensor products and Q-crossed modules. J. London Math. Soc., 51: 243-258.

6:  Baer, R., 1938. Groups with preassigned central and central quotient groups. Trans. Am. Math. Soc., 44: 387-412.

7:  Hall, M. and J.K. Senior, 1964. The Groups of Order 2n (n-6). MacMillan Co., New York, PP: 737-759

8:  Beyl, F.R., U. Felgner and P. Schmid, 1979. On groups occurring as center factor groups. J. Algebra, 61: 161-177.

9:  Beuerle, J., 2005. An elementary classification of finite metacyclic p-groups of clas at least three. Algebra Colloquium. Soc., 43: 651-662.

10:  Rainbolt, J.G. and J.A. Gallian, 2010. Abstract algebra with GAP. Books/Cole Cengage Learning, Boston, http://math.slu.edu/~rainbolt/manual2.html.

11:  Ellis, G., 1998. On the capability of groups. Proc. Edingburgh Math. Soc., 41: 487-495.
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