There are algebraic objects for which inverses are defined, but which aren't groups: namely groupoids, quasigroups and loops. In groupoids the binary operation

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.