GENERAL ⎜ ARTICLE
Combinatorial Group Theory
Group Theory via Generators and Relations
B Sury
B Sury is with the School
of Mathematics, TIFR,
Mumbai.
Group theory revolutionized not only mathematics but
also other sciences. A combinatorial way of describing
groups is by what are called generators and relations. In
this article, our purpose is to discuss this combinatorial
way of describing groups and some of the immediate
applications.
The concept of
groups surfaced
with the
fundamental works
of the great
mathematicians
Evariste Galois
and Niels Henrik
Abel. The advent
of group theory
revolutionized not
only mathematics
but also the other
sciences.
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GENERAL ⎜ ARTICLE
The fact that all
permutations of a
set of objects are
obtainable from
successive
interchanges of
pairs of objects,
can be restated as
saying that a
symmetric group is
generated by its
subset of
transpositions.
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GENERAL ⎜ ARTICLE
1
The nomenclature abelian
is after Abel and is another
word for commutative.
Any (finite) group
can be viewed as
a subgroup of the
group of
permutations of a
(finite) set.
Another way of
viewing a group
is as a quotient
group of a free
group.
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2
Elements x and y of a group
G are said to be conjugate if
there exists z in G such that
x = zyz–1
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What can one say
about a finitely
generated group
where each
element has finite
order? Is such a
group necessarily
finite? This is the
famous Burnside
problem and the
answer is negative
even when the
orders of all the
elements are
bounded by a fixed
number.
3
Work on problems related
to this earned E I Zelmanov
the Fields Medal in 1994.
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GENERAL ⎜ ARTICLE
4
Thus GL(n, ZZ ) is the
collection of all n × n matrices with integer entries and
determinant either +1 or –1.
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GENERAL ⎜ ARTICLE
Suggested Reading
I N Herstein. Topics in
Algebra. Second edition.
Vikas Publishing House,
1976.
For the uninitiated reader, who
wants to look up standard
notation and terminology
used in elementary group
theory, this book is a good
source.
D L Johnson. Presentation
of groups. London Math.
Soc. Lecture Note Series.
No.22. Cambridge University Press.
Address for Correspondence
B Sury
School of Mathematics
Tata Institute of Fundamental
Research, Homi Bhabha Road
Mumbai 400 005, India
email: [email protected]
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