group 241

Gregory series (Leibniz’s series) Named after the

Scottish astronomer and algebraist J

AMES

G

REGORY

(1638–75), the M

ACLAURIN SERIES

for arctan(x) is

sometimes called the Gregory series:

This expression is valid for –1 ≤x≤1. Placing x= 1

into the formula yields the following remarkable for-

mula for

PI

:

Often the name “Gregory series” is used to mean this

particular expression.

See also

INVERSE TRIGONOMETRIC FUNCTIONS

;

M

ADHAVA OF

S

ANGAMAGRAMMA

; T

AYLOR SERIES

.

group Research in pure mathematics is motivated

by one question: what makes mathematics work the

way it does? By identifying the key principles that

underpin one type of mathematical system—be it

geometry and symmetry, or numbers and arithmetic—

mathematicians establish connections between dis-

parate fields: known facts about any one system

satisfying a set of basic principles translate immedi-

ately to analogous facts about a second system satisfy-

ing analogous principles. For example, a study of

arithmetic shows that the operation of addition satis-

fies the same four basic principles as multiplication.

Thus any known fact about addition is accompanied

by a known fact about multiplication (and this corre-

sponding result about multiplication consequently

requires no new proof.)

Motivated by the workings of arithmetic, mathe-

maticians define a group to be any set, often denoted

G, whose elements can be combined in some way that

mimics the addition or the multiplication of the inte-

gers. Specifically, if we denote the result of combining

the elements aand bof Gby the symbol a*b, then Gis

a group if the following four axioms hold:

1. Closure: For all aand bin G, the element a*b is

also a member of G.

2. Associativity: For all a,b,c in Gwe have: a*(b*c) =

(a*b)*c.

3. Existence of an identity: There is an element ein G

so that a*e = a= e*a for all ain G.

4. Existence of inverses: For any ain Gthere is an ele-

ment bin Gsuch that a*b = eand b*a = e.

To honor the work of Norwegian scholar N

IELS

H

ENRIK

A

BEL

(1802–29), mathematicians call G

“Abelian” if a fifth axiom also holds:

5. Commutativity: For all aand bin Gwe have: a*b

= b*a.

These axioms do indeed capture the working prin-

ciples behind both addition and multiplication. For

example, interpreting “*” as addition with “e” as the

number zero, the set of all integers satisfies the axioms

of an Abelian group. (In this setting the inverse of an

integer ais usually denoted –a.) The set of all real num-

bers with zero removed forms an Abelian group under

multiplication. In this context, * is interpreted as the

product operation, “e” is the number 1, and the inverse

of an element ais the number 1/a.

A group could be abstract. For example, the set of

four elements G= {e,a,b,c} is an Abelian group under

an operation * given as follows:

The set of elements G= {1, –1, i, – i, j, – j, k, –k} with

group operation *, given by multiplication as

QUATER

-

NION

s, is an example of a non-Abelian group. The set

of all 2 ×2 invertible matrices with real entries is also

a non-Abelian group under the operation of

MATRIX

multiplication.

GROUP THEORY

is the study of the general structure

of groups and all the results that follow from the four

(or five) basic axioms. The subject is incredibly rich, and

many mathematicians today devote their entire research

careers to the further development of this topic.

Group theory has profound applications to phy-

sics and science. Any physical system that possesses

*eabc

eeabc

aaecb

bbcea

ccbae