Group of Glauberman type for a prime
The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Contents
Definition
Definition for a general finite group
Suppose is a finite group and
is a prime number. We say that
is of Glauberman type with respect to
if the ZJ-functor is a characteristic p-functor whose normalizer generates whole group with p'-core. In the discussion below,
denotes the subgroup obtained by applying the ZJ-functor to
. The ZJ-functor is defined as the center of the Thompson subgroup
, which in turn is defined as the join of abelian subgroups of maximum order.
Explicitly, the following equivalent conditions are satisfied:
- For one (and hence every)
-Sylow subgroup
of
,
: Here,
denotes the
-core of
,
- For one (and hence every)
-Sylow subgroup
of
, the image of
in the quotient
is a normal subgroup of
.
- For one (and hence every)
-Sylow subgroup
of
,
is a normal subgroup of
.
- For one (and hence every)
-Sylow subgroup
of
,
is a characteristic subgroup of
.
Definition for p'-core-free finite group
Suppose is a finite group and
is a prime number. Suppose the p'-core
is trivial, i.e.,
has no nontrivial normal subgroup of order not divisible by
. Then,
is termed a group of Glauberman type for
if it satisfies the following equivalent conditions:
- For one (and hence every)
-Sylow subgroup
,
is a normal subgroup of
.
- For one (and hence every)
-Sylow subgroup
,
is a characteristic subgroup of
.
Equivalence of definitions and its significance
Further information: Equivalence of definitions of group of Glauberman type for a prime
It turns out that, for a finite group and prime number
:
is a group of Glauberman type for
is a group of Glauberman type for
This can be used to provide an alternative definition of group of Glauberman type.
Relation with other properties
Stronger properties
- p-nilpotent group
- strongly p-solvable group (for an odd prime
): For full proof, refer: strongly p-solvable implies Glauberman type for odd p
- Group that is both p-stable and p-constrained for an odd prime
: For full proof, refer: p-constrained and p-stable implies Glauberman type for odd p
Weaker properties
- Group in which the ZJ-functor controls fusion for a prime: For full proof, refer: Glauberman type implies ZJ-functor controls fusion
Examples
p'-core-free examples
In these example groups, is normal in
for one (and hence every)
-Sylow subgroup
of
.
Group | Order | Prime of interest | ![]() |
![]() |
Comment |
---|---|---|---|---|---|
symmetric group:S3 | 6 | 3 | A3 in S3 | A3 in S3 | Here, ![]() ![]() |
alternating group:A4 | 12 | 2 | V4 in A4 | V4 in A4 | Here, ![]() ![]() |
On the other hand, symmetric group:S4 is not a group of Glauberman type for the prime 2.