3 ms·

Actually, isn't having unique inverses a weaker requirement than being a group? Maybe I am wrong...

• crntaylor 14y ago
  There are algebraic objects for which inverses are defined, but which aren't groups: namely groupoids, quasigroups and loops. In groupoids the binary operation isn't total, i.e. there can be pairs (a,b) for which a <> b is not defined -- so we can't use groupoids. In quasigroups and loops the binary operation doesn't need to be associative. But we require associativity, so that we can deduce (a <> b) <> inv b == a <> (b <> inv b) == a <> e == a So we do actually need all the properties of a group.
  • crntaylor 14y ago
    A commenter on Reddit pointed out that quasigroups require both left and right division operators \ and /, and that even though they are not associative, the axioms x \ (x * y) = y x * (x \ y) = y (x * y) / y = x (x / y) * y = x must hold -- so in fact, quasigroups are good enough!