Editing Modular representation theory (section)

== Example ==

Finding a representation of the [[cyclic group]] of two elements over '''F'''<sub>2</sub> is equivalent to the problem of finding [[matrix (mathematics)|matrices]] whose square is the [[identity matrix]].  Over every field of characteristic other than 2, there is always a [[basis (linear algebra)|basis]] such that the matrix can be written as a [[diagonal matrix]] with only 1 or −1 occurring on the diagonal, such as

:<math>
\begin{bmatrix}
1 & 0\\
0 & -1
\end{bmatrix}.
</math>

Over '''F'''<sub>2</sub>, there are many other possible matrices, such as

:<math>
\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}.
</math>

Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the [[Jordan normal form]]. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.