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]