Group

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Group
6 ]5 y1 b/ Q" j3 y/ B, C6 nA group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
, X8 `9 v& j& ]& ^) i/ N to form well-defined products and which furthermore satisfy the following conditions:
1. Closure: If and are two elements in , then the product is also in .
9 N+ U. c. m* x2. Associativity: The defined multiplication is associative, i.e., for all , .
6 u' G' @7 \# y. Q( o: v; m1 U3. Identity: There is an Identity Element
! z9 k5 J* g/ B& b. Q* ~ (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 .
+ T9 E, G7 d( H0 Q7 C( o2 A8 V% @A group is therefore a Monoid
  P/ K5 K2 K5 s. Y1 d for which every element is invertible. A group must contain at least one element.
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The study of groups is known as Group Theory
5 E' U; t) N" S( B6 m7 S( ]. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
of the group.

7 v4 B" P7 t* l+ bSince each element , , , ..., , and is a member of the group, group property 1 requires that the product

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(1)

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must also be a member. Now apply to ,
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" `$ S+ k& {6 l) n- L- I

(2)

2 d$ P* R4 j  M. XBut
9 y+ T" I  B: [& E* s) q

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1 X" B' F6 q8 f- t	(3) 0 T5 L  w! k$ u6 q; w

so

(4) ! }9 z4 ^+ K! I" h- s% p

) V5 Z" K5 r8 Z+ S1 X

which means that

! l: D& d4 ]+ l, U

(5)

) i4 Z( ?: X8 n, U2 W! r

: _( n  K( R# S4 [6 H$ `and

(6)

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