An important result with many applications in the theory of finite groups is the following:
Let
S
≠
1
S \not =1
be a finite p-group for some prime p. Then
S
S
contains a characteristic subgroup
W
(
S
)
≠
1
W(S) \not = 1
with the property that
W
(
S
)
W(S)
is normal in every finite group
G
G
of characteristic
p
p
with
S
∈
S
y
l
p
(
G
)
S \in Syl_p(G)
that does not possess a section isomorphic to the semi-direct product of
S
L
2
(
p
)
SL_{2}(p)
with its natural module.
For odd primes, this was first established by Glauberman with his celebrated
Z
J
ZJ
-Theorem.
The case
p
=
2
p=2
proved more elusive and was only established much later by the second author. Unlike the
Z
J
ZJ
-Theorem, it was not possible to give an explicit description of the subgroup
W
(
S
)
W(S)
in terms of the internal structure of
S
S
.
In this paper we introduce the notion of “almost quadratic action” and show that in each finite p-group
S
S
there exists a unique maximal elementary abelian characteristic subgroup
W
(
S
)
W(S)
which contains
Ω
1
(
Z
(
S
)
)
\Omega _1(Z(S))
and does not allow non-trivial almost quadratic action from
S
S
. We show that
W
(
S
)
W(S)
is normal in all finite groups
G
G
of characteristic p with
S
∈
S
y
l
p
(
G
)
S \in Syl_p(G)
provided that
G
G
does not possess a section isomorphic to the semi-direct product of
S
L
2
(
p
)
SL_{2}(p)
with its natural module. This provides a unified approach for all primes
p
p
and gives a concrete description of the subgroup
W
(
S
)
W(S)
in terms of the internal structure of
S
S
.