Group of Glauberman type for a prime

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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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Definition

Definition for a general finite group

Suppose G is a finite group and p is a prime number. We say that G is of Glauberman type with respect to p if the ZJ-functor is a characteristic p-functor whose normalizer generates whole group with p'-core. In the discussion below, Z(J(P)) denotes the subgroup obtained by applying the ZJ-functor to P. The ZJ-functor is defined as the center of the Thompson subgroup J(P), which in turn is defined as the join of abelian subgroups of maximum order.

Explicitly, the following equivalent conditions are satisfied:

  1. For one (and hence every) p-Sylow subgroup P of G, G = O_{p'}(G)N_G(Z(J(P))): Here, O_{p'}(G) denotes the p'-core of G,
  2. For one (and hence every) p-Sylow subgroup P of G, the image of Z(J(P)) in the quotient G/O_{p'}(G) is a normal subgroup of G/O_{p'}(G).
  3. For one (and hence every) p-Sylow subgroup Q of K = G/O_{p'}(G), Z(J(Q)) is a normal subgroup of K.
  4. For one (and hence every) p-Sylow subgroup Q of K = G/O_{p'}(G), Z(J(Q)) is a characteristic subgroup of K.

Definition for p'-core-free finite group

Suppose G is a finite group and p is a prime number. Suppose the p'-core O_{p'}(G) is trivial, i.e., G has no nontrivial normal subgroup of order not divisible by p. Then, G is termed a group of Glauberman type for p if it satisfies the following equivalent conditions:

  1. For one (and hence every) p-Sylow subgroup P, Z(J(P)) is a normal subgroup of G.
  2. For one (and hence every) p-Sylow subgroup P, Z(J(P)) is a characteristic subgroup of G.

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 G and prime number p:

G is a group of Glauberman type for p \iff G/O_{p'}(G) is a group of Glauberman type for p

This can be used to provide an alternative definition of group of Glauberman type.

Relation with other properties

Stronger properties

Weaker properties

Examples

p'-core-free examples

In these example groups, Z(J(P)) is normal in G for one (and hence every) p-Sylow subgroup P of G.

Group Order Prime of interest p-Sylow subgroup Z(J(P)) Comment
symmetric group:S3 6 3 A3 in S3 A3 in S3 Here, P = Z(J(P)) and it itself is normal in G.
alternating group:A4 12 2 V4 in A4 V4 in A4 Here, P = Z(J(P)) and it itself is normal in G.

On the other hand, symmetric group:S4 is not a group of Glauberman type for the prime 2.