The group theoretical term for what is known to physicists, by way of its connection with matrix traces, as the
trace. The powerful group orthogonality
theorem gives a number of important properties about the structures of groups,
many of which are most easily expressed in terms of characters. In essence, group
characters can be thought of as the matrix traces
of a special set of matrices (a so-called irreducible
representation) used to represent group elements and whose multiplication corresponds
to the multiplication table of the group.

All members of the same conjugacy class in the same representation have the same character. Members of other conjugacy
classes may also have the same character, however. An (abstract) group
can be identified by a listing of the characters of its various representations,
known as a character table. However, there exist
nonisomorphic groups which nevertheless have the same character
table, for example
(the symmetry group of the square) and (the quaternion group).