ISSN 0001-4346, Mathematical Notes, 2012, Vol. 92, No. 5, pp. 636–642. © Pleiades Publishing, Ltd., 2012.
Published in Russian in Matematicheskie Zametki, 2012, Vol. 92, No. 5, pp. 699–706.
Gr ¨ obner–Shirshov Bases for Extended Modular,
Extended Hecke, and Picard Groups
*
E. G. Karpuz
1**
and A. S. ¸ Cevik
2***
1
Karamano ˘ glu Mehmetbey University, Turkey
2
Sel ¸ cuk University, Turkey
Received December 12, 2008
Abstract—In this paper, Gr ¨ obner–Shirshov bases (noncommutative) for extended modular, ex-
tended Hecke and Picard groups are considered. A new algorithm for obtaining normal forms of
elements and hence solving the word problem in these groups is proposed.
DOI: 10.1134/S0001434612110065
Keywords: extended modular group, extended Hecke group, Gr ¨ obner–Shirshov bases, word
problem.
1. INTRODUCTION AND PRELIMINARIES
Algorithmic problems such as the word, conjugacy and isomorphism problems have played an
important role in group theory since the work of M. Dehn in the early 1900’s. These problems are called
decision problems, they require a “yes” or “no” answer to a specific question. Among these decision
problems, the word problem in groups and semigroups has been studied most widely (see [1]). It is well
known that the word problem for finitely presented groups is not solvable in general; that is, given any
two words expressed in the generators of the group, there may be no algorithm to decide whether these
words represent the same element in this group.
The method of Gr ¨ obner–Shirshov bases, which is the main topic of this paper, gives a new algorithm
for obtaining normal forms of elements of groups/semigroups. Therefore, we have a new algorithm for
solving the word problem in these groups/semigroups. Having this in mind, our aim in this paper is to
find Gr ¨ obner–Shirshov bases for extended modular, extended Hecke, and Picard groups.
(a) Gr
¨
obner–Shirshov basis. The theories of the Gr ¨ obner and Gr ¨ obner–Shirshov bases were
invented independently by Shirshov [2] (for noncommutative and nonassociative algebras), Hironaka [3],
and Buchberger [4] (for commutative algebras). The technique of Gr ¨ obner–Shirshov bases has proved
to be very useful in the study of presentations of associative algebras, Lie algebras, semigroups, groups,
and Ω-algebras by considering generators and relations (see, for example, the book [5] by Bokut and
Kukin, and the survey papers [6], [7]).
Let X be a set, let X
∗
be the set of X-words (monomials), and let < be a monomial well ordering
of X
∗
(i.e., < is a well ordering that respects left and right multiplications by words). Also, let k be a field,
and let kX be the free algebra over X and k. For f ∈ kX, let f be the maximal (leading) monomial
of f . Then fg = f g for any f and g. The polynomial f is said to be monic if the coefficient at f in f
is equal to 1. Thus, for monic polynomials f and g, and a, b ∈ X
∗
the Gr ¨ obner–Shirshov basis can be
defined as follows:
(I) Let w be a word such that
w = fb = a g with deg( f ) + deg( g) > deg(w).
∗
The text was submitted by the authors in English.
**
E-mail: eylem.guzel@kmu.edu.tr
***
E-mail: sinan.cevik@selcuk.edu.tr
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