Nonempty associative quasigroup equals group
Contents
History
The fact that a (nonempty) associative quasigroup is the same thing as a group is more than a mere curiosity. Many of the initial definitions of group, including Cayley's first attempted definition in 1854 and Weber's definition in 1882, defined a group as an associative quasigroup, without explicitly mentioning identity elements and inverses.
Further information: History of groups
Statement
Let be a nonempty magma (set with binary operation) satisfying the following two conditions:
-
is a quasigroup under
, i.e., for any
, there exist unique
such that
-
is associative, or
is a semigroup under
, i.e., for any
, we have
Then is a group under
.
Conversely, any group is an associative quasigroup. Associativity is part of the definition, and the quasigroup part is also direct. For full proof, refer: Group implies quasigroup
Related facts
- Nonempty alternative quasigroup equals alternative loop: Here, alternativity, which is a weaker condition than associativity, also forces a nonempty quasigroup to have a neutral element.
Facts used
Proof
Associativity is given to us, so we need to find an identity element (neutral element for the multiplication) and inverses.
Finding the identity element (or neutral element)
Since is nonempty, we can pick
. By the quasigroup condition, find
such that
.
Now, pick any . By the quasigroup condition again, there exists a unique element
such that
. Plugging in, we get:
Thus, is a right neutral element (or right identity) for the multiplication
.
By a similar procedure, we can find a left neutral element. Further, since any left and right neutral element are equal, we see that must be a two-sided identity element for
.
Finding inverses
If is the identity element, then for any
, we can find
such that
. Thus,
has both a left inverse and a right inverse. By associativity, any left and right inverse must be equal, so
has a two-sided inverse. Thus, every element has a two-sided inverse, so every element is invertible, and
is thus a group.