Editing Multiplication (section)

==Multiplication in group theory==<!--linked from below-->
There are many sets that, under the operation of multiplication, satisfy the axioms that define [[group (mathematics)|group]] structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.

A simple example is the set of non-zero [[rational numbers]]. Here identity 1 is had, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, zero must be excluded because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example, an [[abelian group]] is had, but that is not always the case.

To see this, consider the set of invertible square matrices of a given dimension over a given [[field (mathematics)|field]]. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the [[identity matrix]]) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.

Another fact worth noticing is that the integers under multiplication do not form a group—even if zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1 and&nbsp;−1.

Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements). So multiplying element '''a''' by element '''b''' could be notated as '''a''' <math>\cdot</math> '''b''' or '''ab'''. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by <math>\left( \mathbb{Q}/ \{ 0 \} ,\, \cdot \right)</math>.<ref>{{cite book |last1=Burns |first1=Gerald |title=Introduction to group theory with applications |date=1977 |publisher=Academic Press |location=New York |isbn=9780121457501 }}</ref>