Equivalence of normality and characteristicity conditions for isomorph-free p-functor

Statement

Let $p$ be a prime number. Suppose $W$ is a characteristic p-functor that always returns an isomorph-free subgroup of its input group. Then, the following are equivalent for a finite group $G$:

1. For every pair of $p$-Sylow subgroups $P,Q$ of $G$, $W(P)=W(Q)$.
2. For every pair of $p$-Sylow subgroups $P,Q$ of $G$, $W(P)\leq Q$.
3. Each of these:
   • There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)\leq O_{p}(G)$ where $O_{p}(G)$ is the p-core of $G$.
   • For every $p$-Sylow subgroup $P$ of $G$, $W(P)\leq O_{p}(G)$ where $O_{p}(G)$ is the $p$-core of $G$.
4. Each of these:
   • There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)$ is a characteristic subgroup of $G$.
   • For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a characteristic subgroup of $G$.
5. Each of these:
   • There exists a $p$-Sylow subgroup $P$ of $G$ such that $W(P)$ is a normal subgroup of $G$.
   • For every $p$-Sylow subgroup $P$ of $G$, $W(P)$ is a normal subgroup of $G$.

Related facts

• Equivalence of normality and characteristicity conditions for conjugacy functor: This is more general, because any isomorph-free p-functor is weakly closed, but the converse need not hold.
• Equivalence of definitions of weakly closed conjugacy functor: This is a somewhat more powerful version of the statement, albeit it has different hypotheses and conclusions. It can really be thought of as a local version.

Facts used

1. Sylow implies order-conjugate
2. Isomorph-free implies intermediately characteristic
3. Characteristicity is transitive
4. Characteristic implies normal
5. Equivalence of definitions of p-core

Proof

Fact (1) shows that the For every versions are equivalent to the there exists versions. This proves the equivalence of the two versions of (3), the two versions of (4), and the two versions of (5). The remaining directions are each individually quite easy and are summarized below.

We prove the equivalence of (1) with (2). Within this the (2) implies (1) direction requires the use of isomorph-free. We prove the equivalence of (2) with (3). This is straightforward. We then cyclically prove the equivalence of (3), (4) and (5). Of this cyclic proof, the (3) implies (4) part uses isomorph-free via Facts (2) and (3).

From	To	Facts used	Given data used	Explanation
Condition (1)	Condition (2)	--	--	Immediate from the fact that $W(Q)\leq Q$ on account of $W$ being a characteristic $p$-functor.
Condition (2)	Condition (1)		$W$ is isomorph-free	If $W(P)\leq Q$, then we know that $W(P),W(Q)$ are isomorphic subgroups of $Q$. By the isomorph-free condition, we must have $W(P)=W(Q)$.
Condition (2)	Condition (3)	--	--	If $W(P)\leq Q$ for all $p$-Sylow subgroups $Q$, we have $W(P)\leq \bigcap _{Q\in \operatorname {Syl} _{p}(G)}Q=O_{p}(G)$.
Condition (3)	Condition (2)	--	--	$W(P)\leq O_{p}(G)$ and $O_{p}(G)\leq Q$ for all $p$-Sylow subgroups $Q$ forces $W(P)\leq Q$.
Condition (3)	Condition (4)	Facts (2),(3)	$W$ is isomorph-free	We have $W(P)\leq O_{p}(G)\leq P\leq G$. By Fact (2), $W(P)$ is characteristic in $O_{p}(G)$. Separately, $O_{p}(G)$ is characteristic in $G$. Fact (3) now gives that $W(P)$ is characteristic in $G$.
Condition (4)	Condition (5)	Fact (4)	--	Fact-direct
Condition (5)	Condition (3)	Fact (5)		By definition of p-core, $W(P)$ being normal forces $W(P)\leq O_{p}(G)$.