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# Group Character

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).

Character Table, Conjugacy Class, Finite Group, Group, Group Orthogonality Theorem, Matrix Trace

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## References

James, G. D. and Liebeck, M. Representations and Characters of Groups. Cambridge, England: Cambridge University Press, 1993.

Group Character

## Cite this as:

Weisstein, Eric W. "Group Character." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GroupCharacter.html