The idea of an inverse operation that undoes the effect of a given operation is one that is encountered in many different contexts. [In arithmetic we will encounter the inverse operations of addition and subtraction, multiplication and division, raising to a power and calculating a root or logarithm. In calculus the processes of differentiation and integration.]
An operation O in a set S is permutative if the equation x O y = z has a unique solution for each of x, y and z, given the other two within the set, and S is termed a permutoid (or quasi-group) with respect to O. This implies that S is a groupoid with respect to O, since given x and y, z is determined.
The body of the operation table for a permutative operation is a latin square, that is it has the same elements, all different, in every rank and file. A groupoid is permutative if and only if the left and right operators (hO and Oh) determined by each element h are permutations of the elements of the groupoid.
In a permutoid every element is a factor of every other element and has a uniquely defined left cofactor and right cofactor. (If x O y = z then x and y are factors of z and x is the left cofactor of y in z, and y is the right cofactor of x.)
In a permutoid we can define operations |O and O| called the right and left inverse operations of O, such that x O y = z if and only if x = z |O y , so (x O y) |O y = x and ||O = O, and y = x O| z, so x O| (x O z) = z, and O|| = O. Similarly from the commute O we can form the right and left cross-inverses |O an O|. Each of these six operations is permutative.
Theorem. Every element of a commutative permutoid has an exact square root if and only if the system is of odd order. Proof: Let u be a fixed element. If x O y = u then y O x = u, but since u occurs n times in the body of the table there are n/2 pairs (x, y) with x ≠ y such that x O y = u = y O x. If n is odd this leaves one element z such that z O z = u. [A. Sade Ann Soc Sci Bruxelles Ser I, 74 (1960) 91–99.]
A set with an operation that is permutative and associative, i.e. an associative permutoid, is called a group.
Theorem. A group contains a unique identity. Proof. By permutativity the equations e O a = a and a O f = a are uniquely soluble for e and f for any given element a. If x is any other element then e O x = e O (a O k) for some k (by permutativity) = (e O a) O k (by associativity) = a O k (by definition of e) = x (by definition of k). Therefore e is a left identifier for all elements. Similarly f is a right identifier for all elements. Therefore e = f, as has been proved for any groupoid with right and left identifiers.
Theorem. Every element in a group has a unique inverse. Proof: Given an element a we have a O b = e and c O a = e for some b and c by permutativity. We then have b = e O b = (c O a) O b = c O (a O b) = c O e =c.
Theorem. An associative groupoid in which there is an identity and every element has an inverse is a group. This is known as the classical definition of a group. Proof: The solution of x O y = z given y and z is x = z O y* and given x and z is y = x* O z, where * denotes inverse.
Theorem. The classical axioms can be weakened to postulate only the existence of
at least one left identifier and at least one left inverse for every element relative to
that identifier. Proof:
(1) a O a* = e O (a O a*) (by definition of the identifier e) = (a** O a*) O (a O a*) (by definition of inverse) = a** O (a* O (a O a*)) (by associativity) = a** O ((a* O a) O a*) (by associativity) = a** O (e O a*) (by definition of inverse) = a** O a* (by definition of identity) = e (by definition of inverse). This shows a* is also a right inverse.
(2) a O e = a O (a* O a) (by definition of inverse) = (a O a*) O a (by associativity) = e O a (by the right inverse property just proven) = a (by definition of identity). This proves that e is also a right identifier.
(3) Suppose b O a = e then b = b O e = b O (a O a*) = (b O a) O a* = e O a* = a*. This proves the uniqueness of the inverse.
Theorem. For any element a** = a. Proof: a** = e O a** = (a O a*) O a** = a O (a* O a**) = a O e = a.
Theorem. For any elements a and b we have (a O b)* = b* O a*. Note the reversal of order. More generally we may prove that (a O ... O k)* = k* O ... O a*. Proof: (a O b) O (b* O a*) = a O (b O (b* O a*)) = a O ((b O b*) O a*) = a O (e O a*) = a O a* = e.
Theorem. In a group the identity is the only idempotent element. Proof: If a O a = a then a = a O e = a O (a O a*) = (a O a) O a* = a O a* = e.
Theorem. If every element of a group is its own inverse, then the group is commutative. Proof: ???
Theorem. In a group with an even number of elements there is some element, other than the identity, that is its own inverse. Proof: ???
A subset H of a group G that is itself a group with respect to the same operation is called a subgroup. The identity in a subgroup is the same as in the group. A nonempty subset H of G is a subgroup if and only if it is closed with respect to O and *. The intersection of subgroups is a subgroup.
A group is cyclic if it contains at least one element, a, such that all the elements can be expressed in the form a°n. Any cyclic group is commutative. Any subgroup of a cyclic group is cyclic.
A group is sequenceable if its elements can be arranged into a sequence a, b, c, ..., k such that the O-combinations a, a O b, a O b O c, ... a O b O c O ... O k are all distinct. The operation table of a sequenceable group is a complete latin square, that is one in which for any ordered pair of elements (a, b) there exists a rank and file in which a and b appear as adjacent elements in succession.
A (finite) commutative group is sequenceable if and only if it is the direct product of two groups A and B such that A is a cyclic group of order 2^k with k > 0 and B is of odd order. [B. Gordon Pacific J Math 11 (1961) 1309–13]