Proper velocity by Hermann Minkowski's flat-space metric equation . Here a reference map frame of yardsticks and synchronized clocks define map position x and map time "t" respectively, and the "d" preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper velocity where as usual coordinate velocity . Thus finite "w" ensures that "v" is less than lightspeed "c". By grouping "γ" with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass. The proper velocity
Product of group subsets "ST", then "ST" coincides with the semidirect product of "S" and "T". Finally, if both "S" and "T" are normal in "ST", then "ST" coincides with the direct product of "S" and "T". If "S" and "T" are subgroups whose intersection is the trivial subgroup (identity element) and additionally "ST" = "G", then "S" is called a complement of "T" and vice versa. By a (locally unambiguous) abuse of terminology, two subgroups that intersect only on the (otherwise obligatory) identity are sometimes called disjoint. A question that arises in the case of a non-trivial intersection between a normal subgroup "N"