Maschke's Theorem

If a Matrix Group is reducible, then it is completely reducible, i.e., if the Matrix Group is equivalent to the Matrix Group in which every Matrix has the reduced form

$$\left[{\matrix{D_i^{(1)} & X_i\cr 0 & D_i^{(2)}\cr}}\right],$$

then it is equivalent to the Matrix Group obtained by putting $X_i=0$.

See also Matrix Group

References

Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 49, 1987.