# Completely-reducible matrix group

A matrix group $G$ over an arbitrary fixed field $K$, all elements of which may be reduced by simultaneous conjugation by some matrix over $K$ to block-diagonal form, i.e. to the form $$ X = \left( \begin{array}{cccc} d_1(X) & 0 & \ldots & 0 \\ 0 & d_2(X) & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & d_m(X) \end{array} \right) $$ where $d_i(X)$, $i=1,\ldots,m$, are square matrices, the remaining places being filled by zeros, and each block $d_i(G)$ is an irreducible matrix group. In the language of transformations, a group $G$ of linear transformations of a finite-dimensional vector space $V$ over a field $K$ is said to be completely reducible if any one of the following equivalent conditions is met: 1) Any subspace of $V$ which is $G$-invariant has a $G$-invariant direct complement (cf. Invariant subspace); 2) $V$ is decomposable into the direct sum of minimal $G$-invariant subspaces; or 3) $V$ is generated by the minimal $G$-invariant subspaces. Every finite matrix group $G$ over a field $K$ whose characteristic does not divide the order of $G$ is completely reducible. Every normal subgroup of a completely-reducible matrix group is itself completely reducible.

#### References

[1] | Yu.I. Merzlyakov, "Rational groups" , Moscow (1987) (In Russian) |

[2] | M. Hall, "Group theory" , Macmillan (1959) |

#### Comments

A vector space $V$ over $K$ with a completely reducible matrix group $G$ acting is a completely-reducible module for the group ring $K[G]$.

#### References

[a1] | W. Feit, "The representation theory of finite groups" , North-Holland (1982) |

[a2] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) |

**How to Cite This Entry:**

Completely-reducible matrix group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_matrix_group&oldid=38569