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A '''group''' is a monoid all of whose elements are invertible.  That is, a group is a semigroup with a unit element in which every element has an inverse.  Putting all the pieces together, then, a group <math>\underline{X} = (X, *, e)\!</math> is a set <math>X\!</math> with a binary operation <math>* : X \times X \to X\!</math> and a designated element <math>e\!</math> that is subject to the following three axioms:

A '''group''' is a monoid all of whose elements are invertible.  That is, a group is a semigroup with a unit element in which every element has an inverse.  Putting all the pieces together, then, a group <math>\underline{X} = (X, *, e)\!</math> is a set <math>X\!</math> with a binary operation <math>* : X \times X \to X\!</math> and a designated element <math>e\!</math> that is subject to the following three axioms:

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| width="5%" | '''G1.'''

| width="5%" | '''G1.'''

| width="20%" | (associative)

| width="20%" | (associative)

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It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups.  A system <math>\underline{X} = (X, *)\!</math> is given the adjective ''commutative'' if and only if <math>*\!</math> is commutative.  Commutative group, however, are traditionally called ''abelian groups''.  By way of making comparisons with familiar systems and operations, the following usages are also common.

It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups.  A system X = <X, *> is given the adjective "commutative" if and only if * is commutative.  Commutative groups, however, are traditionally called "abelian groups".  By way of making comparisons with familiar systems and operations, the following usages are also common.

1. When one says that X is "written multiplicatively" it means that a raised dot "." or concatenation is used instead of a star for the LOC.  In this case, the unit element is commonly written as a one "1", while the inverse of an element x is written as "x 1".  The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations.  In the multiplicative idiom, the following definitions of "powers", "cyclic groups", and "generators" are also common.

# One says that <math>\underline{X}\!</math> is ''written multiplicatively'' to mean that a raised dot <math>(\cdot)\!</math> or concatenation is used instead of a star for the LOC.  In this case, the unit element is commonly written as an ordinary algebraic one, <math>1,\!</math> while the inverse of an element <math>x\!</math> is written as <math>x^{-1}.\!</math> The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations.  In the multiplicative idiom, the following definitions of ''powers'', ''cyclic groups'', and ''generators'' are also common.

The "nth power" of x in a semigroup X = <X, .>, for positive integer n, is notated as "xn" and defined as follows.  Proceeding recursively, for n = 1, let x1 = x, and for n > 1, let xn = x(n 1).x.

The "nth power" of x in a semigroup X = <X, .>, for positive integer n, is notated as "xn" and defined as follows.  Proceeding recursively, for n = 1, let x1 = x, and for n > 1, let xn = x(n 1).x.