Given a group G, the algebra CG is a vector space

 CG={suma_ig_i|a_i in C,g_i in G}

of finite sums of elements of G, with multiplication defined by g·h=gh, the group operation. It is an example of a group ring.

For example, when the group is the symmetric group on three letters, S_3, the group ring CS_3 is a six-dimensional algebra. An example of the product of elements is


Modules over CG correspond to complex group representations of G. When G is a finite group then CG is a finite-dimensional algebra.

See also

Algebra, Group, Group Representation, Group Ring, Permutation, Ring

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "CG." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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