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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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A characterization of a class of 2-groups by their defining relations 1

Tatjana GRAMUSHNJAK*

Institute of Mathematics and Natural Sciences, Tallinn University, Narva mnt. 25,10120 Tallinn, Estonia

*Corresponding Author:
Tatjana GRAMUSHNJAK
Institute of Mathematics and Natural Sciences
Tallinn University, Narva mnt. 25,10120 Tallinn, Estonia
E-mail: [email protected]

Received Date: January 20, 2008; Revised Date: April 13, 2008

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Abstract

Let n,m be integers such that n ¸ 3, m > 0 and Ck a cyclic group of order k. All groups which can be presented as a semidirect product (C2n+m × C2n) h C2 are described.

Introduction

All non-Abelian groups of order < 32 are described in [1] (Table 1 at the end of the book). M. Jr. Hall and J. K. Senior [3] have given a fully description of all groups of order image There exist exactly 51 non-isomorphic groups of order 32. Some of them can be presented as a semidirect productimage and some of them as a semidirect product image As a generalization of the first case, in [2] all groups of the form image are described. It turned out that there exist only 17 non-isomorphic groups of this form (for a fixed
n). In this paper we generalize the second case. Namely, we shall describe all finite 2-groups
which can be presented in the form image Clearly, each such group G is given by three generators a, b, c and by the defining relations

image

(1.1)

for some image and imagethe ring of residue classes modulo 2k).

The aim of this paper is to prove

Theorem 1.1. For fixed image and image the number of groups which can be given by relations
(1.1) is

image

All possible values of (p, q, r, s) are given in Propositions 3.1, 3.2, 3.3 if m < n, in 3.1, 3.2 if m = n and in 3.4, 3.5 if m > n.

Main concepts for the proof of Theorem 1.1

Let image be a group given by (1.1). An element c induces an inner automorphismimage of order two (the case image= 1 is also included) of group image

image

Therefore, we have to find all automorphisms of image of order two. The map imageimage induces an endomorphism of groupimage if and only if image (mod image). This endomorphism
is an automorphism, if and only if image (mod 2). This map is an automorphism of order two if and only if (p, q, r, s) satisfy the system

image (2.1)

Our purpose is to solve system (2.1). Note that the two first subsystems of (2.1) imply the
following system modulo 2n

image: (2.2)

The solutions (p, q, r, s) of system (2.2) form a set M which was described in [2]. In [2] the set M was given as the union of disjoined subsets image

Let image be a solution of system (2.2), where image and imageimage denotes the set of all invertible elements ofimage Then p and r can be replaced in (2.1) by

image where image

Now it is easy to see that system (2.1) is equivalent to the system

imageimage (2.3)

where image and image and image (2.4)

Remark, that image means the representative of residue class; moreover, we always can choose image

Because the length of the paper is limited, for most of statements we give only idea of proof.

Solving system

The case

Assume that image Then image and system (2.3) takes the form

image (3.1)

Proposition 3.1. Assume that image and q is odd. Then the solutions (p, q, r, s) of (2.1) are of the form imag where imageand image where imageimage There are exactly image solutions of this form if image exactlyimage solutions if m = 2 and exactly image solutions if m = 1.

Proof. The condition of the proposition, conditions (2.4) and image are satisfied
for solution of (2.2) from the set image.While image we have imageimage Where image and imageif image if image Since image and image (mod 2m), the second congruence of (3.1) holds for every image From
the first congruence of (3.1) we get the value for y. Now let us find the number of solutions of the system (2.1). We have image choices for number image choices for odd number image choices for
number x. For i0 we have z = 4 choices if image choices if m = 2 and z = 1 choice if m = 1. This implies that for the number i we have image choices and the number of solutions of the system is equal to the number of triples imageand image

Proposition 3.2. Assume that image q is even and image Then the
solutions of (2.1) are:

image where image and imageimage and in the case image if image then image in the case image then image where image and image There are exactlyimage solutions of this form if image and image solutions if m = n.

image and image

There are exactly 32 solutions of this form.

Proof. To prove the proposition, by [2] we must consider the following sets of solutions of (2.2):

image

where image Solving system (3.1) for each solution of (2.2) from given sets we get from the
second congruence in (3.1) the condition for y and from the first congruence in (3.1) the values for x. The solutions of system (2.2) belonging to set image vgive us solution 1) of system
(2.1). The solutions of system (2.2) belonging to sets image give solution 2) of
system (2.1).

Proposition 3.3. Assume that image and image are both nonzero even numbers, image is odd imageimage and image Then system (2.1) have solutions only if image and these solutions are image where image and if image then imageimage if image there are image solutions of this form. If image there are image solutions.

Proof. Let us now consider the set image The solutions of system (2.2) from this set have the formimage where image image and image (3.2)

The condition image holds only if image The second congruence of (3.1), i.e

image

holds in the case if k = 0 for every image and in the case if k = 1 it holds for every image Since image the first congruence of (3.1), i.e

image

implies

image (3.3)

Since image this congruence holds if and only if

image

The last condition is stronger than (3.2) and implies image where image is the inverse of the odd number u by modulo imageSince image for v we have image values by modulo image in the form

image where image

It follows from (3.3), that in the case m = 1 we have image and in the case m > 1 we have

image

Calculating the number of all obtained solutions, we get the second statement of proposition

The case m > n

The condition image implies g = 0 and image i.e y is even,image where image System (2.3) has now
the form

image (3.4)

Lemma 3.1. The solution of the congruence

image where image

image

Proof. The solutions of imageimage are image i.e.image if f=1 Then image

Lemma 3.2. The solution of the congruence

image where image and image and imageimage

image

Proof. Denote image then image Using (2.1)–(2.10)
in [2], we get that the solution of the last congruence is

image

where image

Now let us find x. Since image it follows that imageimage Analogously .if image then image and imageimage then image

Denote by x1 solutions from Lemma 3.1 and by x2, z2 solutions from Lemma 3.2.

Proposition 3.4. Assume that m > n and the number q is odd imageThen the solutions of (2.1) are: image then the image where image and imageThere are image solutions of these forms.

Proof. Consider the solutions of system (2.2) belonging to the set M3. The second congruence
of (3.4) holds for every image To solve the first congruence of (3.4), consider two cases for z:image In the first case using Lemma 3.2, we
get solution 1) and in the second case, using Lemma 3.1, we get solution 2).

Proposition 3.5. Assume that m > n, image and both numbers q and g are
even. Then (2.1) have solutions only in case f = i = ±1 and these solutions are:

image

image

image

image

image

where image there are image solutions of these forms.

Proof. Consider solutions of system (2.2) belonging to the sets image Solving system (3.4) and using lemmas 3.1 and 3.2, we get from the set image solutions 1), 2), 3), 4) and from setsimage solution 5). Calculating the number of all obtained
solutions, we get the second statement of the proposition.

Acknowledgement

Research was supported by the Estonian Science Foundation Research Grant 5900, 2004-2007.

References

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