Groups
Group
A group is a set
together with a composition law such that the following properties hold:
- Associativity:
for all . - Identity: there exists
such that for all . - Inverse: for every
there exists such that . We may sometime denote the group by
to emphasize the law of composition.
Remark
Whilst the definition of associativity involves only three elements it implies the generalized associative law, that is, a product of
elements in a group has the same value regardless of any parentheses.
Proposition
Let
be a group. Then
- For any
, the inverse is unique and denoted . - For any
, . . - In
the cancellation laws holds: for , implies that .
Subgroups and Order
Abelian Group
A group is called abelian if the composition law is commutative.
Finite Group
A group is called finite if
is a finite set; otherwise the group is called infinite. The order of a finite group is the number of elements in a finite group, denoted .
Power & Order of an Element
For an element
and in a group and we write for the unambiguous composition of with itself times (this follows from generalized associativity). We also define and . An element has finite order if there is such that ; the minimal such is the order of and denoted . Otherwise we say that has infinite order.
Laws of Exponents
Let
be a group, let , and let and be integers. Then
- If
and commute, then . . .
Subgroup
Let
be a group, a nonempty subset is called a subgroup of if it is a group under the same law of composition of . In such case we write .
Theorem
A nonempty subset
is a subgroup if and only if it is closed under composition and under taking inverses. That is and . The latter two can be consolidated to .
e.g. All subgroups of