P-constrained group

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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

Let G be a finite group and p be a prime number. We say that G is p-constrained if the following is true for one (and hence, any) p-Sylow subgroup of G:

C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G).

Here, C_G(P) denotes the centralizer of P in G. O_{p',p} is the second member of the lower pi-series for \pi = \{ p \}.

Definition for a p'-core-free finite group

This is the same as the previous definition, restricted to p'-core-free groups.

Let G be a finite group and p be a prime number. Suppose further that the p'-core of G is trivial, i.e., O_{p'}(G) is the trivial group. Equivalently, every nontrivial normal subgroup of G has order divisible by p. Then, we say that G is p-constrained if its p-core is a self-centralizing subgroup, i.e.,:

\! C_G(O_p(G)) \le O_p(G)

Equivalence of definitions and its significance

Further information: equivalence of definitions of p-constrained group

It turns out that, from the above definitions:

G is p-constrained \iff G/O_{p'}(G) is p-constrained.

This allows us to define p-constraint for arbitrary finite groups in terms of p-constraint for p'-core-free finite groups.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
strongly p-solvable group
p-solvable group p-solvable implies p-constrained p-constrained not implies p-solvable
finite solvable group (via p-solvable) (via p-solvable)
p-nilpotent group (via p-solvable) (via p-solvable)
finite nilpotent group (via finite solvable) (via finite solvable)

Incomparable properties

Property Meaning Proof of one non-implication Proof of other non-implication
p-stable group p-constrained not implies p-stable p-stable not implies p-constrained
group of Glauberman type for a prime p-constrained not implies Glauberman type Glauberman type not implies p-constrained

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No p-constraint is not subgroup-closed It is possible to have a finite group G, a subgroup H, and a prime number p such that G is p-constrained and H is not.
quotient-closed group property No p-constraint is not quotient-closed It is possible to have a finite group G, a normal subgroup N, and a prime number p such that G is p-constrained and G/N is not.