Equivalence of definitions of group of Glauberman type for a prime
From Groupprops
This article gives a proof/explanation of the equivalence of multiple definitions for the term group of Glauberman type for a prime
View a complete list of pages giving proofs of equivalence of definitions
Statement
Suppose is a finite group and
is a prime number. 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. The following are equivalent:
- 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
.
Facts used
Proof
The proof follows directly from Fact (1), where the characteristic p-functor that we use is the ZJ-functor.