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 crystallography 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 POLYGON forms a group called the nth dihedral group. It
has 2n members consisting of n rotations (including the
identity element) and n reflections. The set of rotations
in and unto itself forms a group of just n elements.
Notice that n is a factor of 2n. In general, any subset H
of a group G that itself satisfies the four axioms of a
group is called a subgroup of G. French mathematician
JOSEPH-LOUIS LAGRANGE (1736–1813) proved that if H
is a subgroup of a group G, then the number of elements in H evenly 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 LINEAR GROUP; HOMOMORPHISM; ISOMORPHISM; RING.
group theory The general study of GROUPs and the
results that follow from the basic axioms that define
them is called group theory. Many of the key principles behind group theory were first identified by the
German mathematician CARL FRIEDRICH GAUSS
(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
ÉVARISTE GALOIS (1811–32); who devised the innovative 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 important consequences in physics.
See also ABSTRACT ALGEBRA; FIELD; GROUP; RING.
242 group theory
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 crystallography 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 POLYGON forms a group called the nth dihedral group. It
has 2n members consisting of n rotations (including the
identity element) and n reflections. The set of rotations
in and unto itself forms a group of just n elements.
Notice that n is a factor of 2n. In general, any subset H
of a group G that itself satisfies the four axioms of a
group is called a subgroup of G. French mathematician
JOSEPH-LOUIS LAGRANGE (1736–1813) proved that if H
is a subgroup of a group G, then the number of elements in H evenly 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 LINEAR GROUP; HOMOMORPHISM; ISOMORPHISM; RING.
group theory The general study of GROUPs and the
results that follow from the basic axioms that define
them is called group theory. Many of the key principles behind group theory were first identified by the
German mathematician CARL FRIEDRICH GAUSS
(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
ÉVARISTE GALOIS (1811–32); who devised the innovative 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 important consequences in physics.
See also ABSTRACT ALGEBRA; FIELD; GROUP; RING.
242 group theory