©Electronic Publishing House
SOME REMARKS ON THE ALGEBRAIC STRUCTURE OF THE FINITE COXETER GROUPF4
MUHAMMAD A. ALBAR and NORAH AL-SALEH (Received 10 October 1996 and in revised form 31 January 1997)
Abstract.We consider in this paper the algebraic structure and some properties of the finite Coxeter groupF4.
Keywords and phrases. Presentation, Reidemeister-Schreier method, Coxeter groups.
1991 Mathematics Subject Classification. 20F05.
1. Introduction. The groupF4is one of the irreducible Coxeter groups [9] defined by the presentation
F4=
x1,x2,x3,x4|xi2=e, 1≤i≤4 x1x23
= x3x43
= x2x34
= x1x32
= x1x42
= x2x42
=e . (1) It has the graph
1 2 3 4
4
It is obvious that the groupB3whose graph is
1 2 3
4
is a subgroup ofF4. The order ofB3is known to be 48 [4]. It is easy to see that the index ofB3inF4is 24 and hence the order ofF4is 1152.
2. The structure ofF4. We defineF4by the presentation given in Section 1. We consider the symmetric group of degree 3 with the presentation
S3=
x,y|x2=y2=(xy)3=e
. (2)
We define the mapθ:F4 →S3, where θ
x1
=x, θ x2
=y, θ x3
=θ x4
=e. (3)
It is easy to see thatθis an epimorphism and soF4/kerθS3. We use the Reidemei- ster-Schreier process to find a partition for kerθ.
82 MUHAMMAD A. ALBAR AND NORAH AL-SALEH A Schreier transversal for kerθinF4is
U=
e,x1,x2,x1x2,x2x1,x1x2x1
. (4)
The process gives us the following partition for kerθ:
kerθ=
a,b,c,d|a2=b2=c2=d2=(ab)2=(bc)2
=(ad)3=(bd)3=(cd)3=(ac)2=e . (5) Therefore, kerθis the Coxeter groupD4whose graph is
a d b
a
This shows that the groupF4is the split extension of the Coxeter groupD4byS3. Remark1. To identify the structure ofD4, we consider the mapθ:D4 →S4, where D4is defined by the graph above andS4is defined by the graph
x y z
andθ(a)=x, θ(d)=y, θ(b)=z, andθ(c)=y. Using the Reidemeister-Schreier pro- cess, we find that kerθZ23. Thus,D4is the split extension ofZ23byS4. An alternative method is given in [3], whereDnis shown to be the semi-direct product ofZ2n−1by Sn.
Remark2. A third method to show thatFD4S3follows. We considerD4and S3as having the following graphs:
x2
x4
x3
x1
x y
wherex=(12)andy=(23). We consider the natural action ofS3orD4defined as x1,x2,x3,x4x
=
x2,x1,x3,x4
and
x1,x2,x3,x4y
=
x1,x3,x2,x4 . (6) We letEto be the split extension ofD4byS3with this action. A presentation forEis
E=
x1,x2,x3,x4,x,y| Relations ofD4,Relations ofS3,Action ofS3onD4 . (7) (See [2].) Simple Tietze transformations show thatEF4. Hence,F4D4S3.
3. The derived series ofF4. We use the Reidemeister-Schreier process several times to find the derived series of F4. Firstly, let F4 have the presentation in Section 1.
F4/F4Z2×Z2and we find thatF4=
x,y|x3=y3=
x−1y−1xy2
=e
. The group F4/F4 Z3×Z3 and we getF4 =
a,b,c,d|a2=b2=c2=d2=(ab)2=(ac)2= (cd)2=(bd)2=(bdca)2=e
. Finally,F4 /F4 Z24 and we findF4 =Z2. Thus, we have proved thatF4is solvable of derived length 4.
4. The center and the growth series ofF4. We have seen in Section 2 thatF4 D4S3and thatD4Z23S4. It is easy to see that the center ofD4isZ2(in general, Z
Dn
=Z2ifnis even and{e}ifnis odd [3]). SinceZ S3
= {e}, we see thatZ F4
⊆ Z
D4
=Z2. LetZ D4
be generated byg. From the Reidemeister-Schreier process, we can findgin terms of the generators ofF4and show that it does not commute with any of them. Hence,Z
F4
= {e}.
The growth series (in the sense of Gromov and Milnor) ofF4is [5]
γ F4
=(1+t)4
1+t22 1+t4
1−t+t22
1+t+t22
1−t2+t4
. (8) The order ofF4is obtained here asγ
F4
(1)=24×22×2×32=1152.
Acknowledgement. The first author thanks King Fahd University of Petroleum and Minerals for the support he has got to conduct this research.
References
[1] N. A. Al Saleh,On the finite Coxeter groups, Ph.D. thesis, College of Girls, Dammam, Saudia Arabia, 1994.
[2] M. A. Albar,On presentation of group extensions, Comm. Algebra12(1984), no. 23-24, 2967–2975. MR 86g:20040. Zbl 551.20017.
[3] M. A. Albar and N. A. Al Saleh,The Coxeter groupDn, submitted.
[4] ,On the affine Weyl group of typeBn, Math. Japon.35 (1990), no. 4, 599–602.
MR 91d:20030. Zbl 790.20048.
[5] M. A. Albar, N. A. Al Saleh, and M. A. Al Hamed,The growth series of Coxeter groups,47 (1998), no. 3, 417–428.
[6] C. T. Benson and L. C. Grove,Finite reflection groups, Bogden & Quigley, Inc., Publishers, Tarrytown on Hudson, N.Y., 1971. MR 52 4099. Zbl 579.20045.
[7] N. Bourbaki,Elements de mathematique. Groupes et algebres de Lie, Actualites Scien- tifiques et Industrielles, no. 1337, Hermann, Paris, 1968 (French), Chapitre IV:
Groupes de Coxeter et systemes de Tits. Chapitre V: Groupes engendres par des reflexions. Chapitre VI: systemes de racines. MR 39#1590. Zbl 186.33001.
[8] N. Broderick and G. Maxwell,The crystallography of Coxeter groups. II, J. Algebra44 (1977), no. 1, 290–318. MR 58 11162b. Zbl 348.20041.
[9] H. S. M. Coxeter and W. O. J. Moser,Generators and relations for discrete groups, fourth ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin,New York, 1980. MR 81a:20001.
Zbl 422.20001.
[10] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Ad- vanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.
MR 92h:20002. Zbl 768.20016.
[11] G. Maxwell, The crystallography of Coxeter groups, J. Algebra 35 (1975), 159–177.
MR 58 11162a. Zbl 312.20029.
[12] M. Suzuki, Group theory. I, Grun1dlehren der Mathematischen Wissenschaften [Fun- damental Principles of Mathematical Sciences], vol. 247, Springer-Verlag, Berlin,
84 MUHAMMAD A. ALBAR AND NORAH AL-SALEH
New York, 1982, Translated from the Japanese by the author. MR 82k:20001c.
Zbl 472.20001.
Albar: Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran,31261, Saudi Arabia
Al-Saleh: Department of Mathematics, College of Girls, Dammam, Saudi Arabia
Special Issue on Space Dynamics
Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities. This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics. It is possible to make a division in two main categories: astronomy and astrodynamics. By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth. Many important topics of research nowadays are related to those subjects.
By astrodynamics, we mean topics related to spaceflight dynamics.
It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the grav- itational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects. Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts. Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines. The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research. All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
Before submission authors should carefully read over the journal’s Author Guidelines, which are located athttp://www .hindawi.com/journals/mpe/guidelines.html. Prospective au- thors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sy- stem athttp://mts.hindawi.com/according to the following timetable:
Manuscript Due July 1, 2009 First Round of Reviews October 1, 2009 Publication Date January 1, 2010
Lead Guest Editor
Antonio F. Bertachini A. Prado,Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, 12227-010 São Paulo, Brazil;[email protected]
Guest Editors
Maria Cecilia Zanardi,São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;
Tadashi Yokoyama,Universidade Estadual Paulista (UNESP), Rio Claro, 13506-900 São Paulo, Brazil;
Silvia Maria Giuliatti Winter,São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;
Hindawi Publishing Corporation http://www.hindawi.com
