Equivalence of definitions of group of Glauberman type for a prime
This article gives a proof/explanation of the equivalence of multiple definitions for the term group of Glauberman type for a prime
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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 .
- Equivalence of definitions of characteristic p-functor whose normalizer generates whole group with p'-core
The proof follows directly from Fact (1), where the characteristic p-functor that we use is the ZJ-functor.