Appendix A
Groups
Groups are abstract mathematical structures characterized
by an operation satisfying certain conditions. A structure < Go > is a group
iff
- o is an operation on set G and
- o is associative, and
- G has a unique identity element, and
- each element of G has unique inverse element.
An identity element is defined with respect to a given operation. If o is an operation on a
set G, then e is the identity element with respect to o iff
![( A x)[(x (- G)((x o e = x) /\ (e o x = x))]](th26x.gif)
For
example for the operation addition, +, 0 is the identity element, and for the
operation multiplication, ×, 1 is the identity element. Inverse elements are defined as
follows: Let x and y be any elements of a set G with operation o and identity element
e. The y is inverse of x iff
For
example (-2 + 2 = 0) where 0 is the identity element for the operation +.
(2 × 1/2 = 1) where 1 is the identity element for the operation multiplication. Given
these definitions we can show that the set of integers Z with the operation + will
form a structure < Z,+ > which is a group, because
- for every x,y
Z((x + y) Z), which means that the set Z is closed
under the operation +;
- for every x,y,z Z(x + (y + z)) = ((x + y) + z), i.e. + is associative;
- there exists an element 0 Z, such that for any x Z(x+0) = (0+x) = 0,
i.e., 0 is the identity element with respect to +; and
- for any x Z there exists an element -x Z such that x + (-x) =
(-x) + x = 0, i.e., -x is inverse of x with respect to +.
The set of integers Z under the operation multiplication ×, does not form a group
because for any element x Z, x, may not have 1/x Z.