Group

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Group
  G6 T4 m( [1 |# \# cA group is defined as a finite or infinite set of Operands
: |0 b. e( q9 `" |. y' _* s( P4 D" a (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
2 Q  a7 Z5 {* \/ {: o 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 .
; H" k) X) M1 Z, T3 K. Z2. Associativity: The defined multiplication is associative, i.e., for all , .
3. 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 .
! Z  t" v, s1 M  m" ^: g9 Z* `A group is therefore a Monoid
9 c* O: o4 ~3 D% {& Q for which every element is invertible. A group must contain at least one element.
3 Z  ?6 K. n, n0 A
9 V$ D% u: e; l2 |The study of groups is known as Group Theory
  l* }2 e7 N. r( @3 N) Y  c. If there are a finite number of elements, the group is called a Finite Group
1 R+ Z( q+ g+ j7 [3 A' W, q and the number of elements is called the Order
of the group.

Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1)

, [9 ~& w& j! h" y: n9 Y! Qmust also be a member. Now apply to ,

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(2) 2 h7 w4 D! A# m" a" p5 ]: S

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/ A) ?9 \+ R& b; c4 ?# mBut

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	(3) ! g* y: Y5 b! ?; [& k) Y) p5 c

so
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(4) / x  f  r0 y; x: p, `/ S* G+ b) ~5 ~" ]

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1 p7 O: Y" G/ d- }2 e; Dwhich means that

(5)

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and
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% [6 N! R  l3 s4 u( t" n

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