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Definition/Summary
A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses.
The number of elements of a group is called its order, and the minimum power of an element that will yield the identity is that element's order. The identity's order is 1, and every other group element's order is greater.
Equations
Associativity: \forall a,b,c \in S ,\ (a \cdot b) \cdot c = a \cdot (b \cdot c)
Identity e: \forall a \in S,\ e \cdot a = a \cdot e = a
Inverse: \forall a \in S,\ \exists a^{-1} \in S,\ a \cdot a^{-1} = a^{-1} \cdot a = e
Extended explanation
The smallest nonempty group is the identity group, with set {e} and operation e*e = e.
As a simple example of what one can do in group theory, let us prove by construction that the equations a*x = b and b*x = a both have unique solutions for x in terms of a and b.
a-1*a*x = e*x = x = a-1*b
and
x*a*a-1 = x*e = x = b*a-1
One can also show that there is exactly one idempotent element, the identity, and no zeros (absorbing elements).
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses.
The number of elements of a group is called its order, and the minimum power of an element that will yield the identity is that element's order. The identity's order is 1, and every other group element's order is greater.
Equations
Associativity: \forall a,b,c \in S ,\ (a \cdot b) \cdot c = a \cdot (b \cdot c)
Identity e: \forall a \in S,\ e \cdot a = a \cdot e = a
Inverse: \forall a \in S,\ \exists a^{-1} \in S,\ a \cdot a^{-1} = a^{-1} \cdot a = e
Extended explanation
The smallest nonempty group is the identity group, with set {e} and operation e*e = e.
As a simple example of what one can do in group theory, let us prove by construction that the equations a*x = b and b*x = a both have unique solutions for x in terms of a and b.
a-1*a*x = e*x = x = a-1*b
and
x*a*a-1 = x*e = x = b*a-1
One can also show that there is exactly one idempotent element, the identity, and no zeros (absorbing elements).
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
