Group

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Group
" D. r3 g2 \2 ^  i, \A group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
8 ]$ e3 |3 y" N7 x5 M7 e' K3 s; ^: g to form well-defined products and which furthermore satisfy the following conditions:
% Q9 y- @/ P9 N1. Closure: If and are two elements in , then the product is also in .
. h- n/ `7 A. N( T1 n8 U' W1 B2. Associativity: The defined multiplication is associative, i.e., for all , .
, k( k3 h8 q3 T# K  ^: Z3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
7 A  U: l' e7 U3 Q8 ^5 b7 ZA group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.
  v& m2 g# h6 Z& f5 }
The study of groups is known as Group Theory
5 o; j- n. k6 t- b. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
& O1 d) f* H8 S8 G8 ~% n of the group.

. o& Z$ _! `, lSince each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1) + f9 U6 A* ^. k

  Z" a, t3 l/ K* J9 Q$ |
7 \' q9 {6 X7 m& f# Kmust also be a member. Now apply to ,

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(2)

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6 s8 X) H% ~$ G/ bBut
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{! `& o; z" K	) [9 P. v% o5 B7 t% T* g
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	(3)

so

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(4) $ [- e' Y9 v2 ?1 U# e6 b0 r

! w+ t. ?: _/ ]* Q" X4 g% h+ Fwhich means that
. V+ c  }3 \2 X

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(5)

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and
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