The isomorphism types of finite Lie primitive subgroups of the complex Lie groups $E_{6} ( \C )$ and $F_4 ( \C )$ are determined.
Here, we call a finite subgroup of a complex Lie group $G$
{\sl Lie primitive\/} if it is not contained in a proper closed
subgroup of $G$ of positive dimension. Induction can be used to
investigate subgroups which are not Lie primitive.
Some additional information is provided, such as the characters of these finite subgroups on some small-dimensional modules for the Lie groups.
In studying these groups,
we mainly use two rational linear representations of
the universal covering group $\widetilde E$ of $E_6(\C)$,
namely a 27-dimensional module (there are two inequivalent ones),
denoted by $\K$, and the adjoint module. In particular, we make heavy
use of the characters of $\widetilde E$ on these modules.
The group $F_4(\C)$ occurs in $\widetilde E$
as the stabilizer subgroup of a vector in $\K$.
The finite simple groups of which a perfect central extension occurs in
$F_4(\C)$ or $E_6(\C)$ are:
via $G_2$: $Alt_5$, $Alt_6$, $L(2,7)$, $L(2,8)$, $L(2,13)$, $U(3,3)$,
via $F_4$: $Alt_7$, $Alt_8$, $Alt_9$, $L(2,17)$, $L(2,25)$, $L(2,27)$,
$L(3,3)$, ${}^3D_4(2)$, $U(4,2)$, $O(7,2)$, $O^{+}(8,2)$,
via $E_6$: $Alt_{10}$, $Alt_{11}$, $L ( 2 , 11)$, $L(2,19)$,
$L(3,4)$, $U(4,3)$, ${}^2F_4(2)'$, $M_{11}$, $J_2$.
This list has been found using the classification
of the finite simple groups. On the basis of this list, the finite Lie primitive
subgroups are found to be either the normalizers of one of these subgroups or of one of the two elementary abelian $3$-groups found by Alekseevskii.
1991 Mathematics Subject Classification: 20K47, 20G40, 17B45, 20C10.