Transversal (combinatorics)
In mathematics, given a collection C of sets, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of C (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal. One variation, the one that mimics the situation when the sets are mutually disjoint, is that there is a bijection f from the transversal to C such that x is an element of f(x) for each x in the transversal. In this case, the transversal is also called a system of distinct representatives. The other, less commonly used, possibility does not require a one-to-one relation between the elements of the transversal and the sets of C. Loosely speaking, in this situation the members of the system of representatives are not necessarily distinct.
In group theory, given a subgroup H of a group G, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. In this case, the "sets" (cosets) are mutually disjoint, i.e. the cosets form a partition of the group.
As a particular case of the previous example, given a direct product of groups , then H is a transversal for the cosets of K.
In general, since any equivalence relation on an arbitrary set gives rise to a partition, picking any representative from each equivalence class results in a transversal.
...
Wikipedia