View source for 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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Merzlyakov,   "Rational groups" , Moscow  (1987)  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  M. Hall,   "Group theory" , Macmillan  (1959)</TD></TR>
</table>

====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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feit,   "The representation theory of finite groups" , North-Holland  (1982)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR>
</table>

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