Group

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Group
  v/ h5 _0 }1 v) R9 {! TA group is defined as a finite or infinite set of Operands
) K+ [* p0 E! I2 A' }& g (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
to form well-defined products and which furthermore satisfy the following conditions:
1 c; I& ^+ k: B2 ^: R1 g) a1. Closure: If and are two elements in , then the product is also in .
+ m4 H/ o$ _8 m: T2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
: f; ]5 g9 S/ _2 l( ^ (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 .
A group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.

6 o2 q9 z' z# q, B) h* ^4 P' }7 H+ ZThe study of groups is known as Group Theory
* ?# O" C, T5 N/ i7 y- m. 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.

" B' O( ~2 E6 k" m5 U+ u& d* RSince each element , , , ..., , and is a member of the group, group property 1 requires that the product

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(1)

7 H3 J" z7 D# [2 Z$ L# B- ^must also be a member. Now apply to ,

( ]* A( {( h) f( W  u

(2)

) W0 ^9 `( M. f* L; c& z- P7 L* I5 vBut
6 V/ [3 I( B1 Z; E9 d

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# B0 V; V( d4 E1 k
0 E% i) o3 W4 d: B3 F	(3) % @3 o) z8 i* b" x5 p" \& E7 T

so

(4)

% h1 Z1 U7 i' s: }$ H" I

. r& U  p8 H  Fwhich means that
# F# S. v) `0 u0 y) J% l) m4 x

(5) . k2 x2 A6 V3 A, t4 C1 u3 g

, p0 T+ T, M4 _% t& D0 k) ^. L$ K' e
% H" J+ y! ^( h2 x
and

+ Z  ]: {' o( ^2 R$ s/ m

(6)

" S  f+ j3 l7 I- L9 f0 c
1 q+ O- D9 w7 K. [5 Q: z8 U" @! Y
1 C6 h" k5 S. z: r

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