Editing Quasigroup (section)

== Examples ==
* Every [[group (mathematics)|group]] is a loop, because {{nowrap|1=''a'' ∗ ''x'' = ''b''}} [[if and only if]] {{nowrap|1=''x'' = ''a''<sup>−1</sup> ∗ ''b''}}, and {{nowrap|1=''y'' ∗ ''a'' = ''b''}} if and only if {{nowrap|1=''y'' = ''b'' ∗ ''a''<sup>−1</sup>}}.
* The [[integer]]s '''Z''' (or the [[rational numbers|rationals]] '''Q''' or the [[real number|reals]] '''R''') with [[subtraction]] (−) form a quasigroup. These quasigroups are not loops because there is no identity element (0 is a right identity because {{nowrap|1=''a'' − 0 = ''a''}}, but not a left identity because, in general, {{nowrap|1=0 − ''a'' ≠ ''a''}}).
* The nonzero rationals '''Q'''<sup>×</sup> (or the nonzero reals '''R'''<sup>×</sup>) with [[division (mathematics)|division]] (÷) form a quasigroup.
* Any [[vector space]] over a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] not equal to 2 forms an [[idempotent]], [[commutative]] quasigroup under the operation {{nowrap|1=''x'' ∗ ''y'' = (''x'' + ''y'') / 2}}.
* Every [[Steiner system|Steiner triple system]] defines an [[idempotent]], [[commutative]] quasigroup: {{nowrap|1=''a'' ∗ ''b''}} is the third element of the triple containing ''a'' and ''b''. These quasigroups also satisfy {{nowrap|1=(''x'' ∗ ''y'') ∗ ''y'' = ''x''}} for all ''x'' and ''y'' in the quasigroup. These quasigroups are known as ''Steiner quasigroups''.{{sfn|ps=|Colbourn|Dinitz|2007|p=497|loc=definition&nbsp;28.12}}
* The set {{nowrap|1={{mset|±1, ±i, ±j, ±k}}}} where {{nowrap|1=ii = jj = kk = +1}} and with all other products as in the [[quaternion group]] forms a nonassociative loop of order 8. See [[hyperbolic quaternion#Historical review|hyperbolic quaternions]] for its application. (The hyperbolic quaternions themselves do ''not'' form a loop or quasigroup.)
* The nonzero [[octonions]] form a nonassociative loop under multiplication. The octonions are a special type of loop known as a [[Moufang loop]].
* An associative quasigroup is either empty or is a group, since if there is at least one element, the [[Quasigroup#Inverse_properties|invertibility]] of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements, thus satisfying all three requirements of a group.
* The following construction is due to [[Hans Zassenhaus]]. On the underlying set of the four-dimensional [[vector space]] '''F'''<sup>4</sup> over the 3-element [[Galois field]] {{nowrap|1='''F''' = '''Z'''/3'''Z'''}} define
*: (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>) ∗ (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>, ''y''<sub>4</sub>) = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>) + (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>, ''y''<sub>4</sub>) + (0, 0, 0, (''x''<sub>3</sub> − ''y''<sub>3</sub>)(''x''<sub>1</sub>''y''<sub>2</sub> − ''x''<sub>2</sub>''y''<sub>1</sub>)).
: Then, {{nowrap|('''F'''<sup>4</sup>, ∗)}} is a [[commutative]] [[Moufang loop]] that is not a group.{{sfn|ps=|Romanowska|Smith|1999|p=93}}
* More generally, the nonzero elements of any [[division algebra]] form a quasigroup with the operation of multiplication in the algebra.