symmetry of some kind, for instance, can be analyzed
through the tools and techniques of this topic. For
example, the six symmetries of an equilateral triangle
form a (non-Abelian) group: there are two rotations, a
clockwise and a counterclockwise rotation of 60°; three
reflections, one about each altitude of the triangle; and
an identity operation corresponding to conducting no
action at all. By declaring the operation “*” to be the
effect of performing one action followed by the other,
one can check that all four group axioms hold for this
system. (The fifth axiom, however, does not hold, since
the performance of rotation followed by a reflection, for
instance, gives a result different from the performance
of that same reflection followed by the rotation.) The
analysis of symmetry plays an important role in crystal-
lography and quantum mechanics. Researchers in these
fields deem group theory an essential tool in their work.
The set of symmetries of any regular n-sided
POLY
-
GON
forms a group called the nth dihedral group. It
has 2n members consisting of nrotations (including the
identity element) and nreflections. The set of rotations
in and unto itself forms a group of just nelements.
Notice that nis a factor of 2n. In general, any subset H
of a group Gthat itself satisfies the four axioms of a
group is called a subgroup of G. French mathematician
JOSEPH
-L
OUIS
L
AGRANGE
(1736–1813) proved that if H
is a subgroup of a group G, then the number of ele-
ments in Hevenly divides the count of elements in G.
This result has interesting consequences when applied
to systems of symmetry, or to groups arising from the
study of
NUMBER THEORY
and
MODULAR ARITHMETIC
.
See also
ABSTRACT ALGEBRA
;
FIELD
;
GENERAL LIN
-
EAR GROUP
;
HOMOMORPHISM
;
ISOMORPHISM
;
RING
.
group theory The general study of
GROUP
s and the
results that follow from the basic axioms that define
them is called group theory. Many of the key princi-
ples behind group theory were first identified by the
German mathematician C
ARL
F
RIEDRICH
G
AUSS
(1777–1855) in his studies of
NUMBER THEORY
and
MODULAR ARITHMETIC
. The development of a group
theory as a subject in its own right, however, is usually
attributed to the young French mathematician
É
VARISTE
G
ALOIS
(1811–32); who devised the innova-
tive tools necessary to study solutions to algebraic
equations in depth and from an abstract perspective.
By identifying the abstract principles that make
algebra and arithmetic work the way they do, group
theory provides a powerful tool for analyzing any
mathematical system that satisfies the same basic
axioms. Applying group theory to the symmetries of a
physical system, for example, can often lead to impor-
tant consequences in physics.
See also
ABSTRACT ALGEBRA
;
FIELD
;
GROUP
;
RING
.
242 group theory