Actions of compact Lie groups on spaces
X
X
with
H
∗
(
X
,
Q
)
≅
Q
[
x
1
,
…
,
x
n
]
/
I
0
{H^{\ast }}(X,{\mathbf {Q}}) \cong {\mathbf {Q}}[{x_1}, \ldots ,{x_n}]/{I_0}
,
Q
∈
I
0
Q \in {I_0}
a definite quadratic form,
deg
x
i
=
2
\deg {x_i} = 2
, are considered. It is shown that the existence of an effective action of a compact Lie group
G
G
on such an
X
X
implies
χ
(
X
)
≡
O
(
|
W
G
|
)
\chi (X) \equiv O(|WG|)
, where
χ
(
X
)
\chi (X)
is the Euler characteristic of
X
X
and
|
W
G
|
|WG|
means the order of the Weyl group of
G
G
. Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree
N
t
(
G
/
T
)
{N_t}(G/T)
is equal to
dim
G
\dim G
if
G
G
is a compact connected Lie group and
T
⊂
G
T \subset G
its maximal torus. Effective actions of compact connected Lie groups
K
K
on
G
/
T
G/T
with
dim
K
=
dim
G
\dim K = \dim G
are completely classified.