Group & c0 ?0 A* Q% z/ q1 pA group is defined as a finite or infinite set of Operands 7 Y: _; i. x. s i1 z5 e. c! j! ` (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator 8 o* \9 l. P4 o to form well-defined products and which furthermore satisfy the following conditions: ! ~* H$ P0 Y1 l. _
1. Closure: If and are two elements in , then the product is also in . " o' m$ V3 w, T7 m" I8 C. E9 E
2. Associativity: The defined multiplication is associative, i.e., for all , . % p8 D) I* V, R; T" j3. Identity: There is an Identity Element0 b) Y/ r- c8 W2 R m9 s
(a.k.a. , , or ) such that for every element . / U6 H: r$ K! V& f
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 . * h! E$ R, `( g/ U2 |0 FA group is therefore a Monoid 8 C1 [! E$ r/ g/ D. s for which every element is invertible. A group must contain at least one element. ) w) {1 X- F1 d- q* P4 ^: e5 Y& W & w: }& q0 s" ^1 i% M3 ^9 ~
The study of groups is known as Group Theory & A/ t: L+ @4 m. W- m7 I. If there are a finite number of elements, the group is called a Finite Group 9 U4 z$ R1 Q& A and the number of elements is called the Order # y2 @! N4 r/ c9 j9 K4 z of the group. " ?8 n! o3 n' p- q3 c* c/ n , m& _9 \+ ^3 R& ]
Since each element , , , ..., , and is a member of the group, group property 1 requires that the product : n L. p# o$ u; b. O
, c7 R+ O4 x: P6 m
(1) U- R" B8 l; }5 ]5 L
+ M' Q# W* R" m* G- S6 Q \ x1 e / m$ k1 {% p1 ?* j- L3 ~( e" T" X# f& t$ ~% y: K
must also be a member. Now apply to , . v% O+ \7 Q" J2 Y( s- }0 W