# Collective Variable Free Transition Path Sampling with Generative Flow
Network

Kiyoung Seong1 Seonghyun Park1 Seonghwan Kim2
Woo Youn Kim2 Sungsoo Ahn1
1POSTECH 2KAIST
{kyseong, shpark26<EMAIL_ADDRESS>
{dmdtka00<EMAIL_ADDRESS>
###### Abstract

Understanding transition paths between meta-stable states in molecular systems
is fundamental for material design and drug discovery. However, sampling these
paths via molecular dynamics simulations is computationally prohibitive due to
the high-energy barriers between the meta-stable states. Recent machine
learning approaches are often restricted to simple systems or rely on
collective variables (CVs) extracted from expensive domain knowledge. In this
work, we propose to leverage generative flow networks (GFlowNets) to sample
transition paths without relying on CVs. We reformulate the problem as
amortized energy-based sampling over molecular trajectories and train a bias
potential by minimizing the squared log-ratio between the target distribution
and the generator, derived from the flow matching objective of GFlowNets. Our
evaluation on three proteins (Alanine Dipeptide, Polyproline, and Chignolin)
demonstrates that our approach, called TPS-GFN, generates more realistic and
diverse transition paths than the previous CV-free machine learning approach.

##  1 Introduction

††Project page: https://anonymous.4open.science/w/tps_gflow/.

In material design and drug discovery, understanding the mechanisms and
kinetics of transitions between meta-stable states of molecular systems, such
as protein folding and chemical reactions [1, 2, 3, 4], is crucial. A
comprehensive study of them requires sampling diverse transition paths [5, 6],
which provide insights into mechanisms and energy landscapes. However,
sampling transition paths through unbiased molecular dynamics (MD) simulations
is challenging due to the high-energy barriers of intermediate states, which
cause an exponential decay in transition probability [7]. Thus, a brute-force
search through unbiased MD simulations is impractical within a realistic
computational budget.

To address this problem, researchers have developed bias potential enhanced
sampling (BPES) methods such as umbrella sampling [8], meta-dynamics [9], and
adaptive biasing force [10, ABF] methods. BPES methods rely on auxiliary bias
potentials (or forces) to facilitate transitions across high-energy barriers.
These methods design bias potentials using collective variables (CVs),
functions of atomic coordinates that capture the slow modes of the transition.
While effective for some systems, the reliance on expensive domain knowledge
limits the applicability of BPES methods to systems where CVs are less
understood.

Recently, machine learning has emerged as a promising paradigm for designing
bias potentials (or forces) without relying on CVs [11, 12, 13, 14]. For
instance, Das et al. [11] and Lelièvre et al. [12] used reinforcement learning
to sample transition paths from biased molecular dynamics with a parametrized
force that approximates the transition path distribution. However, these works
are limited to two-dimensional systems. Holdijk et al. [14] trained the bias
potential by minimizing KL divergence on the trajectories collected by itself,
i.e., on-policy training. Still, this approach suffers from mode-seeking
behavior, and the on-policy training does not reuse the collected
trajectories, leading to inefficiency in sample complexity.

Contribution. In this work, we propose TPS-GFN, generative flow networks [15,
GFlowNets] (GFN) for transition path sampling (TPS), which trains the bias
potential without requiring collective variables (CVs). To this end, we
formulate the problem as the energy-based sampling over molecular trajectories
between the given meta-stable states. GFlowNet allows off-policy training and
avoids collapsing to a particular mode of distribution over transition paths.

In particular, our method trains the bias potential by minimizing the squared
log-ratio between the target distribution and the generator, which we derive
from the one-step trajectory balance (TB) objective [16, 17]. We use a replay
buffer to store the molecular trajectories, which are later used for off-
policy training of the generator. Furthermore, we also design a new reward
function to provide dense training signals and evaluate molecular trajectories
with various lengths between the meta-stable states.

We extensively evaluate our method for three peptides, i.e., Alenine
Dipeptide, Polyproline, and Chignolin. We demonstrate that our method
faithfully generates a realistic transition path between the given meta-stable
states in Figure 1, outperforming the prior ML-based approach [14]. We also
show that our method discovers diverse transition states.

Figure 1: Visualization of transition paths generated by our TPS-GFN,
progressing from left to right over time. (Top) Polyproline transformation
from the meta-stable region PP-II to PP-I. (Bottom) Chignolin folding process.

##  2 Related work

###  2.1 Collective variable-based transition path sampling

Bias potential methods. Bias potential (or force) was originally introduced in
enhanced sampling to explore molecular conformations that are difficult to
access by the unbiased molecular dynamics (MD) within limited simulation times
[18]. We focus on adaptive bias simulations such as meta-dynamics [9] and
adaptive biasing force (ABF) [10], which updates the bias potential (or force)
adaptively. Meta-dynamics and ABF add bias potential (or force) in a time-
dependent manner and enhance the sampling along a few well-designed collective
variables (CVs). Other methods for sampling transition paths using bias
potentials (or forces) include the umbrella sampling [8], and the Wang-Landau
method [19].

MCMC-based methods. Shooting is a well-established method using the Markov
chain Monte Carlo (MCMC) procedure on path space [20]. To be specific, it
perturbs a shooting point in the transition path, performs MD simulations from
it, and proposes new paths. Borrero et al. [21] perturbed the shooting point
with meta-dynamics to avoid traps in path space, and Jung et al. [22] trained
the parameterized commitor function to select the shooting point with a high
acceptance rate. Plainer et al. [23] used latent proposals to leverage
information from the latent space of the Boltzmann generator. However, the
MCMC-based method suffers from long mixing times to sample independent
transition paths, and there is a trade-off between the acceptance rate and
diversity. Falkner et al. [24] generated independent shooting points from the
Boltzmann generator using the bias potential with the umbrella sampling, but
this method relies on CVs and shooting region.

Automated learning of CV. Since many algorithms rely on CVs which require
expensive domain knowledge about transitions, researchers have explored
automating the derivation of CV from data [25, 26, 27, 28]. Bonati et al. [25]
and Trizio et al. [26] used time-lagged independent component analysis and
applied deep linear discriminant analysis on MD trajectories. Ray et al. [27]
and Yang et al. [28] considered the transition path ensemble as effective data
augmentation strategies for discriminant analysis to discover CVs,
respectively.

###  2.2 Collective variable-free transition path sampling

Generation from a fixed dataset. Given a dataset of transition paths, one can
train generative models using the dataset to sample a transition path.
Petersen et al. [13] and Lelièvre et al. [12] applied diffusion probabilistic
models and [29] variational auto-encoders [30], respectively. However, these
methods are limited to small systems, as collecting the transition paths is
notoriously difficult due to high energy barriers.

Bias potential-based method. Without any previously collected dataset of
transition paths, Das et al. [11] and Hua et al. [31] considered transition
path sampling as a reinforcement learning (RL) problem and trained the bias
potentials (or forces) of the policy by minimizing KL divergence. However,
these works are limited to two-dimensional systems. Notably, Holdijk et al.
[14] applied the path integral cross-entropy [32] used in stochastic optimal
control framework to scale up to larger systems.

###  2.3 Generative flow networks

Generative flow networks (GFlowNets) are the learning framework for amortized
inference, generating objects through a series of decisions with a stochastic
policy, or forward policy. [15, 16]. By learning the policy to sample an
object proportionally to their rewards, GFlowNets can produce diverse samples
[15]. To be specific, they train the forward policy by matching its trajectory
flow with an auxiliary backward trajectory flow. Training objectives such as
flow matching [15], trajectory balance [17], and detailed balance [16] have
been proposed. Recent studies have successfully applied GFlowNets to tasks
including molecular conformer generation [33], drug discovery [34], crystal
structure design [35], and biological sequence design [36].

Comparison with RL and variational inference. RL and variational inference
algorithms share similarities with GFlowNets in learning a target distribution
specified by a reward function. GFlowNets and RL both make sequential
decisions to achieve high rewards, but GFLowNets differ by sampling
proportionally to the reward rather than maximizing it. Moreover, while
variational inference algorithms aim to learn the target (Boltzmann)
distribution, GFlowNets reduce gradient variance by estimating marginal
quantities and allow off-policy learning without the need for reweighted
importance sampling [37].

##  3 Method

Figure 2: The potential energy landscape of Alanine Dipeptide with its
collective variable (CV), i.e., two dihedral angles $(\phi,\psi)$. We aim to
generate a transition path from one meta-stable state to another, highlighted
with yellow arrows.

###  3.1 Problem setup

At a high level, our goal is to sample transition paths from one meta-stable
state to another meta-stable state through unbiased molecular dynamics (MD),
e.g., transition paths from the meta-stable region $C5$ to $C7_{ax}$ in
Alanine Dipeptide, as illustrated in Figure 2. However, since such events
occur rarely in naïve MD simulation. To this end, we aim to train a neural
network to search for such transition paths.

Molecular dynamics. We consider MD simulations that describe motion of a
molecular state $\bm{x}=(\bm{r},\bm{v})$ with $N$ atoms, given the atom-wise
positions (or conformation) $\bm{r}\in\mathbb{R}^{N\times 3}$ and the atom-
wise velocities $\bm{v}\in\mathbb{R}^{N\times 3}$. In particular, we consider
Langevin dynamics [38] to describe the molecular motion:

$\mathrm{d}\bm{r}=\bm{v}\mathrm{d}t,\qquad\mathrm{d}\bm{v}=\frac{-\nabla_{\bm{r}}U(\bm{r})}{\bm{m}}\mathrm{d}t-\gamma\bm{v}\mathrm{d}t+\sqrt{\frac{2\gamma
k_{B}\lambda}{\bm{m}}}\mathrm{d}\bm{w},$

where $U(\bm{r})$, $\bm{m}$, $\gamma$, $k_{B}$, $\lambda$, and
$\mathrm{d}\bm{w}$ denote the potential energy, atom-wise masses, the friction
term, Boltzmann constant, temperature, and the Brownian motion, respectively.

Euler Maruyama discretization. Langevin dynamics can be discretized into a
sequence of molecular states $\bm{x}_{0},\bm{x}_{1},\ldots,\bm{x}_{T}$ using
the Euler Maruyama method [39] as follows:

$\bm{r}_{t}=\bm{r}_{t-1}+\bm{v}_{t-1}\Delta,\qquad\bm{v}_{t}=(1-\gamma\Delta)\bm{v}_{t-1}-\frac{\nabla_{\bm{r}}U(\bm{r}_{t})}{\bm{m}}\Delta+\sqrt{\frac{2\gamma
k_{B}\lambda}{\bm{m}}}\bm{\varepsilon}_{t}\Delta,$

where $\Delta$ is the discretization step size and
$\bm{\varepsilon}_{t}\in\mathbb{R}^{N\times 3}$ is a noise from the standard
normal distribution. We denote the conditional probabilities described by this
discretization as $p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})$.

Transition path sampling. One of the challenges in sampling transition paths
through unbiased MD simulations is the meta-stability: a state can remain
trapped for a long time in the initial meta-stable region
$\mathcal{A}\subseteq\mathbb{R}^{N\times 3}$ before transitioning into a
distinct meta-stable region $\mathcal{B}\subseteq\mathbb{R}^{N\times 3}$. Our
goal is to sample a transition path
$\bm{x}_{0:T}=(\bm{x}_{0},\ldots\bm{x}_{T})$ that links the distinct regions
$\mathcal{A}$ and $\mathcal{B}$ where $\bm{r}_{0}\in\mathcal{A}$,
$\bm{r}_{T}\in\mathcal{B}$ and $T$ is the length of the trajectory.

Since the meta-stable region $\mathcal{A}$ and $\mathcal{B}$ are not well-
specified for many molecular systems, we find unique local minima
$\bm{r}_{\mathcal{A}}$ and $\bm{r}_{\mathcal{B}}$ of the potential function
inside the meta-stable regions $\mathcal{A}$ and $\mathcal{B}$, respectively.
Afterwards, we sample a transition path $\bm{x}_{0:T}$ that starts from the
state $\bm{x}_{0}=\bm{x}_{\mathcal{A}}$ and ends nearby the conformation
$\bm{r}_{\mathcal{B}}$, i.e., $\bm{r}_{T}\in\mathcal{B}$. To be specific, we
fix $\bm{x}_{0}=\bm{x}_{\mathcal{A}}$ and sample the remaining trajectory
$\bm{x}_{1:T}$ from the following distribution:

$p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})=\frac{1}{Z}\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})$
(1)

where $Z$ is the normalizing constant and
$\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})$ is the binary indicator function for
the final conformation $\bm{r}_{T}$ being in the meta-stable region
$\mathcal{B}$. In other words, $\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})=1$ if
$\bm{r}_{T}\in\mathcal{B}$, and $\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})=0$
otherwise.

To sample transition paths from the target distribution
$p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})$, one can use rejection
sampling. This involves sampling $\bm{x}_{1:T}$ through unbiased MD
simulations and then checking if $\bm{r}_{T}\in\mathcal{B}$. However, this
method is computationally expensive due to the low acceptance rate, which is
caused by the high-energy barriers.

###  3.2 TPS-GFN: GFlowNets for transition path sampling

In this section, we explain our approach, called TPS-GFN, that amortizes the
expensive cost of sampling transition paths from the distribution
$p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})$. Our key idea is
threefold: (1) using the trajectory balance (TB) objective to train a
generator
$p_{\theta}(\bm{x}_{1:T}|\bm{x}_{0})=\prod_{t=1}^{T}p_{\theta}(\bm{x}_{t}|\bm{x}_{t-1})$
towards the true distribution
$p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})$, (2) relaxation of the
target distribution, and (3) parameterization of the generator with the bias
potential. We provide a full description of the algorithm in Algorithm 1.

Training with trajectory balance (TB). We aim to make the generator
$p_{\theta}(\bm{x}_{1:T}|\bm{x}_{0})$ imitate the target distribution
$p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})$. From Equation 1, we
can achieve this by satisfying the following identity:

$\left(\log\frac{Z\prod_{t=1}^{T}p_{\theta}(\bm{x}_{t}|\bm{x}_{t-1})}{\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})}\right)^{2}=0,$

for all trajectories $\bm{x}_{1:T}$. To this end, we propose to use the one-
step TB objective [17], the squared log-ratio between the unnormalized
generator $Z_{\theta}p_{\theta}(\bm{x}_{1:T}|\bm{x}_{0})$ and the unnormalized
target distribution
$\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})$,
as follows:

$\mathcal{L}_{\text{TB}}(\bm{x}_{1:T};\theta)=\left(\log\frac{Z_{\theta}\prod_{t=1}^{T}p_{\theta}(\bm{x}_{t}|\bm{x}_{t-1})}{\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})}\right)^{2},$
(2)

where $Z_{\theta}\in\mathbb{R}$ is a learnable scalar parameter to estimate
the normalization constant $Z$. We match the generator with the target
distribution by minimizing the loss function
$\mathcal{L}_{\text{TB}}(\bm{x}_{1:T};\theta)$ over trajectories
$\bm{x}_{1:T}$ sampled from an arbitrary training distribution, i.e., off-
policy training. We provide an explicit derivation of the one-step TB
objective (Equation 2) in Appendix A.

To leverage the ability of the TB objective, we use a replay buffer that
collects the past samples from the tempered version of the generator and
reuses them for training. This improve sample efficiency and stability of
training, both for GFlowNets [15] and reinforcement learning [40]. TPS-GFN
alternates between (1) sampling from the generator to store molecular
trajectories in the replay buffer and (2) training the generator on samples
from the replay buffer, as described in Algorithm 1.

Algorithm 1 GFlowNet training for transition path sampling

1: Initialize an empty replay buffer $\mathcal{B}$, the biased potential
$b_{\theta}(\bm{r})$, and the parameter $Z_{\theta}$.

2: for $i=1,\ldots,I$ do

3: Sample $M_{1}$ trajectories $\\{\bm{x}_{0:T}^{(m)}\\}^{M_{1}}_{m=1}$ from
the tempered version of biased MD (Equation 3).

4: Update the replay buffer
$\mathcal{B}\leftarrow\mathcal{B}\cup\\{\bm{x}_{0:T}^{(m)}\\}^{M_{1}}_{m=1}$.

5: for $j=1,\ldots,J$ do

6: Sample $M_{2}$ paths $\\{\bm{x}_{0:T}^{(m)}\\}^{M_{2}}_{m=1}$ from the
replay buffer $\mathcal{B}$.

7: Update parameters $\theta$ to minimize the loss $\mathcal{L}$ (Equation 2)
using the samples $\\{\bm{x}_{0:T}^{(m)}\\}^{M_{2}}_{m=1}$.

8: end for

9: end for

As shown by Malkin et al. [37], the TB objective automatically performs
gradient variance reduction for variational inference by estimating a marginal
quantity (the partition function) that acts as a baseline and allows off-
policy learning without the need for reweighted importance sampling. This is
also a slight modification to the VarGrad objective [41, 42] that was proposed
to reduce the gradient of evidence lower bound (ELBO).

We note the similarity of our TPS-GFN with the prior work [14], which
minimizes the Kullback-Leibler (KL) divergence between the generator and the
target distribution. In comparison, the TB objective yields better training
dynamics due to reduced variance in the gradients, and better sample
efficiency due to the ability to reuse the collected sample with an external
replay buffer. Furthermore, TPS-GFN does not suffer from the mode-seeking
behavior of KL divergence.

Target distribution relaxation. Learning to imitate the target distribution
$p_{\mathcal{A},\mathcal{B}}$ with TB objective is challenging since its
probability is rarely non-zero. To alleviate this issue, we relax the target
distribution by (1) allowing trajectories to reach the meta-stable region
$\mathcal{B}$ before termination and (2) relaxing the binary indicator
function into a smooth function. In particular, we relax the target
distribution into the following form:

$\tilde{p}_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})=\frac{1}{\tilde{Z}}\tilde{\mathbf{1}}_{\mathcal{B}}(\bm{r}_{1:T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1}),$

where $\tilde{Z}$ is the normalizing constant and
$\tilde{\mathbf{1}}_{\mathcal{B}}(\bm{r}_{1:T})$ is the relaxed indicator
function defined as follows:

$\tilde{\mathbf{1}}_{\mathcal{B}}(\bm{r}_{1:T})=\max_{t\in\\{1,\ldots,T\\}}\exp\left(-\frac{\lVert
D(\bm{r}_{t})-D(\bm{r}_{\mathcal{B}})\rVert_{F}^{2}}{2\sigma^{2}}\right).$

Here, $\lVert\cdot\rVert_{F}$ is the Frobeneous norm, $\sigma>0$ is a
hyperparameter to control the degree of relaxation, and
$D(\bm{r})\in\mathbb{R}^{N\times N}$ is a distance matrix with $(i,j)$-th
element denoting the Euclidean distance $\lVert r_{i}-r_{j}\rVert_{2}$ between
the $i$-th and the $j$-th atoms [43].

Parameterizing generator with bias potential. Similar to Holdijk et al. [14],
we model the generator $p_{\theta}(\bm{x}_{1:T}|\bm{x}_{0})$ as the
discretization of the following biased Langevin dynamics:

$\mathrm{d}\bm{r}=\bm{v}\mathrm{d}t,\quad\mathrm{d}\bm{v}=-\frac{\nabla_{\bm{r}}(U(\bm{r})+b_{\theta}(\bm{r}))}{\bm{m}}\mathrm{d}t-\gamma\bm{v}\mathrm{d}t+\sqrt{\frac{2\gamma
k_{B}\lambda}{\bm{m}}}\mathrm{d}\bm{w},$

where $b_{\theta}(\bm{r})$ is the bias potential that guides the Langevin
dynamics towards the meta-stable region $\mathcal{B}$. We discretize the
biased Langevin dynamics as follows:

$\bm{r}_{t+1}=\bm{r}_{t}+\bm{v}_{t}\Delta,\qquad\bm{v}_{t+1}=(1-\gamma\Delta)\bm{v}_{t}+\frac{-\nabla_{\bm{r}}(U(\bm{r}_{t})+b_{\theta}(\bm{r}_{t}))}{\bm{m}}\Delta+\sqrt{\frac{2\gamma
k_{B}\lambda}{\bm{m}}}\bm{\varepsilon}_{t}\Delta,$ (3)

We model the bias potential $b_{\theta}(\bm{r})$ or directly bias force
$\mathbf{b}_{\theta}(\bm{r})=\nabla_{\bm{r}}b_{\theta}(\bm{r})$ for the
generator. We use a multi-layered perception (MLP) architecture to
parameterize the bias potential or force.

##  4 Experiment

Table 1: Benchmark scores on Alanine Dipeptide which have well-known CVs. All
metrics are averaged over 64 trajectories. ETP and EFP are reported only if
the final state is in the target meta-stable state. The best values are
highlighted in bold except for MD. Values from Holdijk et al. [14] are
indicated by $\dagger$. Methods using bias force and potential are denoted by
(F) and (P), respectively.

Method | EPD ($\downarrow$) | THP ($\uparrow$) | ETP ($\downarrow$) | LL ($\uparrow$) | EPCD ($\downarrow$) | EFP ($\downarrow$)
---|---|---|---|---|---|---
$\mathrm{nm}\times 10^{-3}$ | $\mathrm{\%}$ | $\mathrm{kJmol}^{-1}$ | $\times 10^{2}$ | $\mathrm{nm}$ | $\mathrm{kJmol}^{-1}$
MD (300K) | $7.78\pm 0.61$ |  0$0.00$ |  $-$$-$ |  0$-$5.02 $\pm$ 0.15 |  0$88.27\pm 5.65$0 |  0.$-$
MD (9000K) | $7.26\pm 2.47$ |  03.12 |  $-$770.18 $\pm$ 151.85 | $-25.47\pm 4.91$ | $101.82\pm 31.02$ | $\phantom{-}604.02\pm 136.73$
PIPS (F) | $2.06\pm 0.27$ | $56.25$ |  $-$000.48 $\pm$ 10.310 |  0$-$4.49 $\pm$ 0.36 |  0$12.15\pm 3.49$0 | $-15.85\pm 11.21$
PIPS (P)† | $1.21\pm 0.31$ | $63.50$ |  00$-$8.35 $\pm$ 8.0400 |  00.$-$ |  0$-$ |  0.$-$
TPS-GFN (F) | $1.05\pm 0.11$ | 98.43 |  0$-\textbf{18.13 $\pm$ 4.67}$00 |  0$-$4.91 $\pm$ 0.16 |  00$5.12\pm 2.67$0 |  0$-\textbf{45.58 $\pm$ 7.12}$00
TPS-GFN (P) | 1.02 $\pm$ 0.08 | 98.43 |  0$-$16.62 $\pm$ 5.1000 |  0$-$$4.85\pm 0.12$ |  004.91 $\pm$ 1.700 |  0$-45.18\pm 6.88$00

Table 2: Benchmark scores on Polyproline and Chignolin. All metrics are
averaged over 64 trajectories. The best values are highlighted in bold except
for MD. $\ddagger$ indicates reproduced results from the prior work [14].
Methods using bias force and potential are denoted by (F) and (P),
respectively.

Molecule | Method | EPD ($\downarrow$) | LL ($\uparrow$) | EPCD ($\downarrow$)
---|---|---|---|---
$\mathrm{nm\times 10^{-3}}$ | $\times 10^{2}$ | $\mathrm{nm}$
| MD (300K) | $18.83\pm 1.59$ |  $-$00 4.45 $\pm$ 0.57 0 | $82.43\pm 3.69$
| PIPS (F)‡ | $29.13\pm 2.94$ | $-215.82\pm 87.03$ |  $111.46\pm 4.86$ .
Polyproline | PIPS (P)‡ | $12.21\pm 1.77$ | $-210.27\pm 96.11$ | $76.90\pm 6.17$
|  TPS-GFN (F) |  07.94 $\pm$ 2.13 |  00$-6.93\pm 0.98$0 | 42.85 $\pm$ 8.00
|  TPS-GFN (P) | $13.86\pm 1.08$ |  $-$002.64 $\pm$ 0.520 | $83.23\pm 4.59$
| MD (300K) | $381.68\pm 48.74$ |  $-$010.11 $\pm$ 1.030 | $259.63\pm 12.58$
| PIPS (F)‡ |  068.02 $\pm$ 20.63 | $-251.19\pm 63.00$ | $188.56\pm 29.77$
Chignolin | PIPS (P)‡ |  068.87 $\pm$ 9.390 | $-241.22\pm 58.15$ | $186.30\pm 11.79$
|  TPS-GFN (F) |  $\phantom{0}53.34\pm 5.92$0 |  0$-40.02\pm 4.07$0 |  $157.44\pm 6.92$0
|  TPS-GFN (P) |  026.31 $\pm$ 1.810 |  00$-$0.88 $\pm$ 1.420 | 101.43 $\pm$ 7.270

###  4.1 Molecular systems

In this section, we evaluate our TPS-GFN on three molecular systems, (1)
Alanine Dipeptide, (2) Polyproline Helix, and (3) Chignolin. We run molecular
dynamics (MD) simulations on a femtosecond ($\mathrm{fs}$) scale to sample
paths and evaluate them at the temperature of 300K, using the OpenMM library.
We report more details about OpenMM and model configurations in Section B.1
and Section B.2, respectively.

Alanine Dipeptide. The Alanine Dipeptide is a small peptide consisting of two
alanine residues linked by a peptide bond. Alanine Dipeptide has a well-known
collective variable (CV), i.e., two dihedral angles $(\phi,\psi)$. We sample
the transition paths from the meta-stable region $C5$ to $C7_{ax}$ for a 500
$\mathrm{fs}$.

Polyproline Helix and Chignolin. In our experiments, the Polyproline Helix
consists of 3 proline residues while the Chignolin consists of 10 amino acid
residues. Their CVs and mechanisms are less understood than Alanine Dipeptide
due to their complexity. We sample transition paths of Polyporline between the
meta-stable region PP-II and PP-I for a 10000 $\mathrm{fs}$, and Chignolin
folding process for a 5000 $\mathrm{fs}$. In training, we use the tempered
version of biased MD simulations at a temperature of 600K.

(a) Molecular dynamics (MD)

(b) PIPS

(c) TPS-GFN (Ours)

Figure 3: Visualization of 64 paths starting in meta-stable region $C5$ on the
Ramachandran plot of Alanine Dipeptide. White circles indicate meta-stable
states, and stars indicate transition states. (3(a)) The paths from unbiased
MD simulations that fail to escape the initial meta-stable region. (3(b)) The
paths generated by PIPS, passing through only one transition state. (3(c)) The
paths generated by TPS-GFN, passing through both transition states. For
clarity, 10 paths are highlighted.

###  4.2 Quantitative results

In our experiments, we compare our method with unbiased MD and PIPS [14]. To
compare the generated transition paths, we consider three metrics following
the prior work [14]. Expected pairwise distance (EPD) measures the similarity
between the final conformation and the target conformation by comparing their
interatomic distances. Similar to EPD, target hit percentage (THP) evaluates
how many paths achieve the target meta-stable region in a binary manner.
Unlike EPD and THP which only focus on the final conformation of the
transition path, energy transition point (ETP) measures the maximum potential
energy of a transition path, referring to the ability of a model to find the
transition states when crossing the energy barrier.

However, ETP only evaluates the potential energy of a transition state, it can
not reflect the realisticity of an overall transition path. To resolve this
issue, we propose log-likelihood (LL) which refers to log-likelihood from the
unbiased MD simulation. Similarly, since EPD and THP only consider the
distance between the final conformation of sampled paths and the target
conformation, they can not reflect the potential energy of the final
conformation which explains its stability and realisticity. To evaluate it, we
propose the potential energy of the final point (EFP), and the expected
pairwise Coulomb distance (EPCD) which consider both the potential energy of
the final conformation and the distance between the final conformation of
sampled paths and the target conformation. EPCD measures both geometric and
electrostatic similarity between the final conformation and the target
conformation using the Coulomb matrix. The Coulomb matrix captures atomic mass
and energy level around the final state better than the distance matrix. More
details of metrics are in Section B.3.

Alanine Dipeptide. Since Alanine Dipeptide has well-known CVs, i.e., two
dihedral angles $(\phi,\psi)$, we can evaluate THP, ETP, and EFP which
requires defining the meta-stable region with a CV. In Table 1, our TPS-GFN
outperforms the baselines for both previous metrics (EPD, THP, ETP) and
proposed metrics (LL, EPCD, EFP) regardless of predicting bias potential or
force. This shows that the TB objective used in our method leads to more
realistic and high-quality transition paths than KL divergence used in PIPS.
In particular, the $1.02$ EPD and $98.43\%$ THP score of TPS-GFN show that our
method faithfully amortizes the rejection sampling from unbiased MD
simulations. We also highlight that the EFP of our method is much closer to
the potential energy of the target conformation $\bm{r}_{C7_{ax}}$, $-84.86$
$\mathrm{kJmol^{-1}}$, than the baselines.

Polyproline Helix and Chignolin. Unlike Alanine Dipeptide, the CVs of
Polyproline Helix, and Chignolin are less understood. Therefore, we exclude
metrics THP, ETP, and EFP which require CVs to define the meta-stable state.
In Table 2, our method consistently outperforms baselines in both Polyproline
Helix and Chignolin, regardless of predicting bias potential or force. Our
model samples more realistic transition paths than the baselines with respect
to the overall path and final conformation. We note that as the size of the
molecule is larger, the EPD and EPCD performances decreases. For the large
size of molecules including Chignolin, the bias potential produces better
transition paths than bias force since the output of bias potential is always
scalar regardless of the size of the molecule and the external force from the
bias potential is conservative.

###  4.3 Qualitative results

In this section, we provide a qualitative comparison between our method and
baselines and 3D visualizations. For Alanine Dipeptide, we show that our
method generates diverse transition paths passing through both transition
states. For Polyproline Helix and Chignolin, we visualize the transformation
of molecular systems with their potential energy plot. For more visualization,
please refer to Appendix C, and project page for animation of the generated
transition paths.

Alanine Dipeptide. In Figure 3, we compare transition paths from our method to
paths from baselines using the Ramachandran plot. We sample transition paths
from $C5$ to $C7_{ax}$ which passes one of the two transition states in the
energy landscape indicated by white stars in Figure 3. In this experiment, we
sampled 64 transition paths for all cases. The paths from unbiased MD
simulations in Figure 3(a) remains trapped in the initial meta-stable region,
and thus fail to produce transition paths. Both PIPS [14] and our TPS-GFN
generates valid transition paths between $C5$ and $C7_{ax}$ as shown in Figure
3(b) and Figure 3(c), repectively. However, PIPS generates transition paths
passing through only one of the transition states while our TPS-GFN generates
diverse transition paths, passing through both transition states as shown in
Figure 3(c).

Figure 4: Visualization of a transition path from the meta-stable region PP-II
to PP-I of Polyproline generated by our TPS-GFN. The top row visualizes the 3d
view of states along the time horizon. The bottom row visualizes the potential
energy along the time horizon. The backbone of the Polyproline is highlighted
in green.

Polyproline Helix and Chignolin. Since the CVs of the Polyproline Helix and
the Chignolin are less understood, the Ramachandran plot is unavailable.
Alternatively, we visualize the 3D views of the generated transition path at 5
different time steps with their potential energy along the time horizon. In
Figure 4, Our method successfully generates transition paths in terms of one
CV candidate, handedness [44]. We observe change from the left-handed Helix to
the right-handed Helix of the Polyproline and the potential plot which roughly
increases up to approximately 440 $\mathrm{fs}$ and decreases after then.
Similarly, in Figure 5, our method captures the folding process of the
Chignolin since the potential plot which roughly increases up to approximately
950 $\mathrm{fs}$ and decreases after then. We note that our method can
produce various lengths of transition paths by cutting paths based on the
relaxed indicator function. We also find the average time of the transition
from PP-II to PP-I in Polyproline Helix is approximately 720 $\mathrm{fs}$ and
the average folding time of chignolin is approximately 1000 $\mathrm{fs}$.

Figure 5: Visualization of a folding process of Chignolin generated by our
TPS-GFN. The top row visualizes the 3d view of states along the time horizon.
The bottom row visualizes the potential energy along the time horizon.

##  5 Conclusion

In this work, we introduced a novel CV-free approach to sampling transition
paths using generative flow networks (GFlowNets) to model bias potentials for
efficient and diverse sampling. By reformulating the problem as amortized
energy-based sampling and employing the one-step trajectory balance (TB)
objective, our approach significantly improves the discovery of realistic
transition paths compared to existing methods. Evaluations on proteins such as
Alanine Dipeptide, Polyproline, and Chignolin demonstrate superior accuracy
and diversity in identifying transition states. We believe that our works
expand to more complex systems, with significant implications for drug
discovery and material design.

Limitation. While our algorithm successfully samples transition paths for
small peptides, we did not evaluate our algorithm for larger peptides or
proteins due to the lack of computational budget. In addition, our work does
not generalize across different pairs of meta-stable states or different
molecular systems. This points to an interesting venue for future research,
which would be more appealing for practical applications in drug discovery or
material design.

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##  Appendix A Generative Flow Networks

In this section, we briefly introduce the concept of GFlowNets and explain how
we derive Equation 2. We refer to Lahlou et al. [45] for a comprehensive study
on formalizing GFlowNets on continuous state spaces.

Generative flow networks. GFlowNet [15, 16] is a framework to sequentially
construct samples $s\in\mathcal{S}$ proportional to a reward function $R(s)$,
where $\mathcal{S}$ is a set of data. In other words, GFlowNet aims to learn a
target distribution $p^{\ast}(s)$ defined as:

$p^{\ast}(s)=\frac{R(s)}{Z},\qquad Z=\int_{\mathcal{S}}R(s)ds,$

where $Z$ is the normalizing constant. To this end, GFlowNet samples a
trajectory $\tau=(s_{0},...,s_{L})\in\\{s_{0}\\}\times\mathcal{S}^{L}$ of
length $L$ through a series of state transitions made by a forward policy
$P_{F}(s_{l}|s_{l-1};\theta)$ for $l=1,\ldots,L$, where $s_{0}$ is the fixed
initial state of all trajectories and $s=s_{L}$ is the terminal state we want
to sample.

Flow matching. To train the forward policy, GFlowNet introduces an auxiliary
backward policy $P_{B}(s_{l-1}|s_{l};\theta)$ and matches the forward flow
$Z_{\theta}P_{F}(\tau;\theta)$ with backward flow $R(s)P_{B}(\tau|s;\theta)$
using the flow matching objective over the trajectories
$\tau=(s_{0},...,s_{L})$:

$Z_{\theta}\prod_{l=1}^{L}P_{F}(s_{l}|s_{l-1};\theta)\approx
R(s)\prod_{l=1}^{L}P_{B}(s_{l-1}|s_{l};\theta)$ (4)

where $Z_{\theta}\in\mathbb{R}$ is a learnable scalar parameter to estimate
the normalization constant $Z$. Equation 4 induces $Z_{\theta}P_{F}(s)\approx
R(s)$, by marginalizing forward and backward flows over set of trajectories
$\mathcal{T}(s)$ ending at the terminal state $s$, i.e.,
$\int_{\tau\in\mathcal{T}(s)}Z_{\theta}P_{F}(\tau;\theta)d\tau=Z_{\theta}P_{F}(s)$
and $\int_{\tau\in\mathcal{T}(s)}R(s)P_{B}(\tau|s;\theta)d\tau=R(s)$. In other
words, we can sequentially construct a sample $s$ proportional to a reward
function $R(s)$ through a series of state transitions by the forward policy
after minimizing the flow matching objective.

Trajectory balance. For flow matching, Malkin et al. [17] proposed the
trajectory balance objective:

$\mathcal{L}_{\text{TB}}(s_{1},\ldots,s_{L};\theta)=\left(\log\frac{Z_{\theta}\prod_{l=1}^{L}P_{F}(s_{l}|s_{l-1};\theta)}{R(s_{L})\prod_{l=1}^{L}P_{B}(s_{l-1}|s_{l};\theta)}\right)^{2},$

which corresponds to minimizing the squared difference of logarithms on the
left and right-hand side of Equation 4. In particular, one-step GFlowNets,
i.e., $L=1$, induces the following training objective:

$\mathcal{L}_{\text{TB}}(s_{1};\theta)=\left(\log\frac{Z_{\theta}P_{F}(s_{1}|s_{0};\theta)}{R(s_{1})}\right)^{2},$
(5)

where $P_{B}(s_{0}|s_{1};\theta)$ is ignored since $s_{0}$ is the constant
initial state. Note that the detailed balance [16] objective also reduces to
the same formula as in Equation 5 with one-step GFlowNets.

Transition path sampling via GFlowNets. In this work, we formulate the
transition path sampling problem (Equation 1) as one-step GFlowNets. We set
$\mathcal{S}$ as a set of paths $\bm{x}_{1:T}$ of length $T$ starting at the
initial state $s_{0}=\bm{x}_{0}$ and the reward function as

$R(\bm{x}_{1:T})=\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})\propto
p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0}).$ (6)

Since $P_{F}(s_{1}|s_{0};\theta)=p_{\theta}(\bm{x}_{1:T}|\bm{x}_{0})$, one can
derive Equation 2 from Equation 5:

$\mathcal{L}(\bm{x}_{1:T};\theta)=\left(\log\frac{Z_{\theta}\prod_{t=1}^{T}p_{\theta}(\bm{x}_{t}|\bm{x}_{t-1})}{\mathbf{1}_{\mathcal{B}}(\bm{r}_{T})\prod_{t=1}^{T}p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})}\right)^{2}.$

##  Appendix B Experiment details

We implement codes based on OpenMM [46] and Pytorch [47] libraries, and run
codes on a single GPU (NVIDIA RTX 3090 or RTX A5000).

###  B.1 OpenMM configurations

We run MD simulations to sample paths and evaluate them at the temperature of
300K, using the Langevin integrator. Other configurations depend on molecules
as shown in Table 3.

Table 3: OpenMM configurations depeding on molecules.

Molecule | Time horizon | Time step | Force field | Solvent
---|---|---|---|---
Alanine Dipeptide | 500 $\mathrm{fs}$ | 1 $\mathrm{fs}$ | amber99sbildn | tip3p
Polyproline Helix | 10000 $\mathrm{fs}$ | 2 $\mathrm{fs}$ | amber/protein.ff14SBonlysc | implicit/gbn2
Chignolin | 5000 $\mathrm{fs}$ | 1 $\mathrm{fs}$ | amber/protein.ff14SBonlysc | implicit/gbn2

###  B.2 Model configurations

We report the model configurations used to produce the results from PIPS [14]
and TPS-GFN in Table 4 and Table 5, respectively.

Table 4: Model configurations used to produce the results from PIPS.

Molecule | Rollout | LR | Samples | Brownian motion std
---|---|---|---|---
Alanine Dipeptide | 15000 | $10^{-5}$ | 16 | 0.1
Polyproline Helix | 500 | $3\times 10^{-5}$ | 25 | 0.1
Chignolin | 500 | $10^{-4}$ | 16 | 0.05

Table 5: Model configurations used to produce the results from TPS-GFN.

Molecule | Force | Flexible | Rollout | Trains per rollout | LR | Log Z LR | Samples | Target std($\sigma$)
---|---|---|---|---|---|---|---|---
Alanine Dipeptide | ✗ | ✗ | $10000$ | $2000$ | $0.01$ | $0.001$ | $16$ | $0.05$
Alanine Dipeptide | ✓ | ✗ | $10000$ | $2000$ | $0.01$ | $0.001$ | $16$ | $0.05$
Polyproline Helix | ✗ | ✓ | $5000$ | $200$ | $0.0001$ | $0.01$ | $2$ | $0.2$
Polyproline Helix | ✓ | ✓ | $5000$ | $200$ | $0.001$ | $0.01$ | $2$ | $0.2$
Chignolin | ✗ | ✓ | $5000$ | $200$ | $0.0001$ | $0.01$ | $2$ | $0.2$
Chignolin | ✓ | ✓ | $5000$ | $200$ | $0.0001$ | $0.01$ | $2$ | $0.2$

For Alanine Dipeptide, we remove the flexibility of length in our method by
replacing our relaxed indicator function
$\tilde{\mathbf{1}}_{\mathcal{B}}(\bm{x}_{1:T})$ with
$\tilde{\mathbf{1}}_{\mathcal{B}}(\bm{x}_{T})=\exp\left(-\lVert
D(\bm{r}_{T})-D(\bm{r}_{\mathcal{B}})\rVert_{F}^{2}/2\sigma^{2}\right)$ which
slightly increases performance. However, other molecules benefit from
flexiblity.

###  B.3 Evaluation metrics

Following the prior work [14], we adopt three metrics for evaluation: (i)
Expected pairwise distance (EPD), (ii) Target hit percentage (THP), and (iii)
Energy transition point (ETP).

#### Expected pairwise distance (EPD).

The interatomic pairwise distance matrix $D(\cdot)$ [43] expresses molecular
conformation in an E(3)-invariant manner. EPD measures the similarity between
the final conformation $\bm{r}_{t}$ and the target conformation
$\bm{r}_{\mathcal{B}}$ by calculating the square distance between interatomic
pairwise distance matrices defined as follows:

$\operatorname{EPD}(\bm{r}_{t},\bm{r}_{\mathcal{B}})=\frac{1}{N^{2}}\lVert
D(\bm{r}_{t})-D(\bm{r}_{\mathcal{B}})\rVert_{F}^{2}$

Where $t$ is the index of the final conformation of transition paths.

#### Target hit percentage (THP).

While EPD continuously measures the distance from the fixed target
conformation, THP measures the success of paths arriving at the target meta-
stable state $\mathcal{B}$ in a binary manner. For the final conformations
$\\{r^{(i)}\\}^{M}_{i=1}$ of $M$ paths, THP is defined as follows:

$\operatorname{THP}=\frac{|\\{i:r^{(i)}\in\mathcal{B}\\}|}{M}$

For the alanine dipeptide, we use
$\mathcal{B}=\\{r:|\phi(r)-\phi(r_{\mathcal{B}})|+|\psi(r)-\psi(r_{\mathcal{B}})|<0.75\\}$
where $(\phi,\psi)$ is the collective variable consisting of two torsion
angles.

#### Energy transition point (ETP).

Unlike EPD and THP which evaluate the final conformation of a transition path,
ETP measures the ability of the method to find the transition states when
crossing the energy barrier. ETP refers to the maximum potential energy among
states in a transition path. Formally, given a transition path $\bm{x}_{0:t}$
that reaches the target meta-stable region i.e., $\bm{r}_{t}\in\mathcal{B}$,
ETP computes

$\operatorname{ETP}(\bm{x}_{0:t})=\max_{i\in[0,t]}U(\bm{r}_{i})$

However, we point out that these metrics can not measure the realisticity of
the paths with respect to unbiased molecular dynamics. Thus, we propose
metrics for the realisticity of paths and final conformations. The likelihood
of the unbiased molecular dynamics is a part of the target distribution
$p_{\mathcal{A},\mathcal{B}}(\bm{x}_{1:T}|\bm{x}_{0})$ which explains how
realistic the sampled transition paths are.

#### Log Likelihood (LL).

Log Likelihood (LL) refers to the log-likelihood of unbiased molecular
dynamics as follows:

$\operatorname{LL}(\bm{x}_{0:T})=\frac{1}{T}\sum_{t=1}^{T}\log
p_{\text{MD}}(\bm{x}_{t}|\bm{x}_{t-1})$

We divide by $T$ to evaluate paths regardless of length. For the realisticity
of final conformations, we focus on the potential energy of them. We propose
two metrics, potential Energy of the Final Point (EFP) and Expected Pairwise
Coulomb Distance (EPCD).

#### Potential energy of final point (EFP).

The final conformation of a transition path from unbiased molecular dynamics
should have low potential energy due to meta-stablity. EFP measures the
potential $U(r_{t})$ of the final conformation of a transition path sampled
from the model.

#### Expected pairwise coulomb distance (EPCD).

Coulomb matrix [48] $C(r)$ of the final conformation $r$ describes
electrostatic interaction between nuclei in the final conformation $r$ and is
formulated as follows:

$C(r)_{ij}=\begin{cases}0.5Z_{i}^{2.4}&\text{for }i=j\\\ Z_{i}Z_{j}/\lVert
r_{i}-r_{j}\rVert&\text{for }i\neq j\end{cases}$

where $Z_{i}$ and $Z_{j}$ are the atomic numbers and $r_{i}$, $r_{j}$ are the
positions of atoms $i$ and $j$, respectively. The atomic numbers $Z_{i}$ and
$Z_{j}$ of the Coulomb matrix reflect atomic mass and energy level around the
final conformation better than the distance matrix. We propose EPCD which
replaces the distance metrics used in EPD with the Coulomb matrix and is
defined as follows:

$\operatorname{EPCD}(\bm{r}_{t},\bm{r}_{\mathcal{B}})=\frac{1}{N^{2}}\lVert
C(\bm{r}_{t})-C(\bm{r}_{\mathcal{B}})\rVert_{F}^{2}$

EPCD measures both geometric and electrostatic similarities between the final
conformation $\bm{r}_{t}$ and the target conformation $\bm{r}_{\mathcal{B}}$.

##  Appendix C Additional visualizations

In this section, we provide additional visualizations of transition paths for
each molecular system, generated by our TPS-GFN.

###  C.1 Alanine Dipeptide

In Figure 6, we visualize a collective variable (CV), i.e., two dihedral
angles $(\phi,\psi)$ along the time horizon and 3D views of a transition path
from the meta-stable region $C5$ to $C7_{ax}$ with the target conformation
$\bm{r}_{C7_{ax}}$. Our model successfully generates a transition path since
the final frame in Figure 6(l) is in $C7_{ax}$.

(a) Visualization of $\psi$ and $\phi$ along the time horizon

(b) Frame 0 $(=\bm{r}_{C5})$

(c) Frame 50

(d) Frame 100

(e) Frame 150

(f) Frame 200

(g) Frame 250

(h) Frame 300

(i) Frame 350

(j) Frame 400

(k) Frame 450

(l) Frame 498

(m) $\bm{r}_{C7_{ax}}$

Figure 6: 3D view of a transition path from the meta-stable region $C5$ to
$C7_{ax}$ with the target conformation $\bm{r}_{C7_{ax}}$

###  C.2 Polyproline Helix

In Figure 7, we visualize 3D views of a transition path from the meta-stable
region PP-II to PP-I.

(a) Frame 0 $(=\bm{r}_{\text{PP-II}})$

(b) Frame 40

(c) Frame 80

(d) Frame 120

(e) Frame 160

(f) Frame 200

(g) Frame 240

(h) Frame 280

(i) Frame 320

(j) Frame 357

(k) $\bm{r}_{\text{PP-I}}$

Figure 7: 3D view of a transition path from the meta-stable region PP-II to
PP-I with the target conformation $\bm{r}_{\text{PP-I}}$

###  C.3 Chignolin

In Figure 8, we visualize 3D views of a Chignolin folding process.

(a) Frame 0 $(=\bm{r}_{\text{unfolded}})$

(b) Frame 100

(c) Frame 200

(d) Frame 300

(e) Frame 400

(f) Frame 500

(g) Frame 600

(h) Frame 700

(i) Frame 800

(j) Frame 900

(k) Frame 992

(l) $\bm{r}_{\text{folded}}$

Figure 8: 3D visualization of a Chignolin folding process with the target
conformation $\bm{r}_{\text{folded}}$


# Sharpness of saturated fusion systems on a Sylow $p$-subgroup of ${\rm
G}_{2}(p)$

Valentina Grazian  Department of Mathematics and Applications
University of Milano – Bicocca
Via Roberto Cozzi 55, 20125 Milano
Italy<EMAIL_ADDRESS>and  Ettore Marmo  Department of
Mathematics and Applications
University of Milano – Bicocca
Via Roberto Cozzi 55, 20125 Milano
Italy<EMAIL_ADDRESS>
###### Abstract.

We prove that the Díaz-Park’s sharpness conjecture holds for saturated fusion
systems defined on a Sylow $p$-subgroup of the group ${\rm G}_{2}(p)$, for
$p\geq 5$.

##  1\. Introduction

Given a finite group $G$, a homology decomposition of the classifying space
$BG$ for $G$ is a process of glueing together classifying spaces of a suitable
collection of subgroups of $G$ in order to obtain a space that is isomorphic
to $BG$ in mod-$p$ homology. In [Dwy98] Dwyer proved that the collection of
$p$-centric subgroups of $G$ gives rise to a _sharp_ homology decomposition,
yielding a particularly simple formula for the cohomology of the classifying
space $BG$ in terms of the cohomology of the spaces involved in it:

######  Theorem (Subgroup sharpness for $p$-centric subgroups).

Let $p$ be a prime, let $G$ be a finite group and let $\mathcal{O}_{p}^{c}(G)$
denote the $p$-centric orbit category of $G$. Then $H^{j}(-;\mathbb{F}_{p})$
is acyclic as a controvariant functor over $\mathcal{O}_{p}^{c}(G)$:

${\lim\limits_{\longleftarrow}}_{\mathcal{O}_{p}^{c}(G)}^{i}H^{j}(-;\mathbb{F}_{p})=0$

for all $i\geq 1,j\geq 0.$

Such a context can be extended to saturated fusion systems $\mathcal{F}$,
thanks to the homotopy theory of fusion systems introduced in [BLO03] and the
existence of a unique object that plays the role of the classifying space
$B\mathcal{F}$ for $\mathcal{F}$ ([Che13]). In this view, in [DP15] Díaz and
Park stated the following sharpness conjecture for fusion systems, not only
passing from finite groups to saturated fusion systems, but also replacing the
cohomology functor $H^{j}(-;\mathbb{F}_{p})$ with a more general Mackey
functor $M$ (See Definition 2.3):

######  Conjecture (Sharpness for fusion systems).

Let $p$ be a prime, let $\mathcal{F}$ be a saturated fusion system on a finite
$p$-group $S$, let $\mathcal{O}(\mathcal{F}^{c})$ denote the centric orbit
category of $\mathcal{F}$ and let
$M=(M^{*},M_{*}):\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
be a Mackey functor over the full orbit category $\mathcal{O}(\mathcal{F})$ of
$\mathcal{F}$. Then the contravariant part $M^{*}$ over
$\mathcal{O}(\mathcal{F}^{c})$ is acyclic:

${\lim\limits_{\longleftarrow}}_{\mathcal{O}(\mathcal{F}^{c})}^{i}M^{*}|_{\mathcal{O}(\mathcal{F}^{c})}=0$

for all $i>0$.

In particular, the sharpness conjecture implies the acyclicity of the
cohomology functor
$H^{j}(-;\mathbb{F}_{p})\colon\mathcal{O}(\mathcal{F}^{c})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
for any $j\geq 0$, which is one of the open problems listed in [AKO11, III.7].

As a direct consequence of Dwyer’s subgroup sharpness theorem, the conjecture
holds for the cohomology functor
$H^{j}(-;\mathbb{F}_{p})\colon\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
and any fusion system $\mathcal{F}=\mathcal{F}_{S}(G)$ that is realized by a
finite group $G$ containing $S$ as a Sylow $p$-subgroup. In [DP15] the authors
showed that in fact all fusion systems realized by finite groups satisfy the
sharpness conjecture for any Mackey functor $M$. Therefore it remains to test
the conjecture on the so called exotic fusion systems, that is, saturated
fusion systems not realizable by finite groups. Using the information on
saturated fusion systems defined on a $p$-group having a maximal subgroup that
is abelian uncovered by Oliver([Oli14]), Díaz and Park proved that the
sharpness conjecture also holds in this scenario (see [DP15, Theorem C]). In
this paper, we apply Díaz and Park’s methodology to saturated fusion systems
defined on a Sylow $p$-subgroup of the group ${\rm G}_{2}(p)$, for $p\geq 5$,
to prove that the sharpness conjecture holds for this family of saturated
fusion systems as well.

######  Theorem A.

Let $p\geq 5$ be a prime, let $S$ be a Sylow $p$-subgroup of the group ${\rm
G}_{2}(p)$, let $\mathcal{F}$ be a saturated fusion system on $S$ and let
$M=(M^{*},M_{*}):\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
be a Mackey functor over $\mathcal{O}(\mathcal{F})$. Then

${\lim\limits_{\longleftarrow}}_{\mathcal{O}(\mathcal{F}^{c})}^{i}M^{*}|_{\mathcal{O}(\mathcal{F}^{c})}=0$

for all $i>0$.

Indeed, Theorem A is a consequence of the following stronger result involving
simple Mackey functors $S_{Q,V}$ (Definition 2.6):

######  Theorem B.

Let $p\geq 5$ be a prime, let $S$ be a Sylow $p$-subgroup of the group ${\rm
G}_{2}(p)$ and let $\mathcal{F}$ be a saturated fusion system on $S$. Suppose
$Q\leq S$ is not $\mathcal{F}$-centric and $V$ is a simple
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(Q)$-module. Then for all
$\mathcal{F}$-centric subgroups $P,R\leq S$ such that their intersection
$T=P\cap R$ is not $\mathcal{F}$-centric the composition

(1) $S_{Q,V}(P)\xrightarrow{{\rm Res}_{T}^{P}}S_{Q,V}(T)\xrightarrow{{\rm
Ind}_{T}^{R}}S_{Q,V}(R)$

is zero.

One of the main ingredients used in the proof of Theorem B is the knowledge of
the structure of the $\mathcal{F}$-essential subgroups of $S$, first described
in [PS18] and then in more generality in [GP22], while characterizing
saturated fusion systems defined on a $p$-group having maximal nilpotency
class.

We remark that in a recent paper [HLL22], Henke, Libman and Lynd proved that
the families of fusion systems affording the structure of punctured groups,
such as the Benson-Solomon family [AKO11, 5, III.6.7], the Parker-Stroth
family [PS15] and the Clelland-Parker family [CP10], satisfy the sharpness
conjecture (although they gave an explicit proof only for the cohomology
functor and not for a generic Mackey functor). The sharpness conjecture for
fusion systems is still open for other families of exotic fusion systems.

Organization of the paper. In Section 2 we recall the definitions of orbit
category, Mackey functors and simple Mackey functors for fusion systems and we
prove some basic properties. In Section 3 we work under the assumption that
$\mathcal{F}$ is a saturated fusion system on a finite $p$-group $S$ and $Q,P$
and $R$ are subgroups of $S$ for which the composition of maps (1) is non-
zero. We underline that the results proved in this section hold for any finite
$p$-group $S$ and can represent the starting point for the proof of the
sharpness conjecture for other families of saturated fusion systems. In
Section 4 we specialize our investigation on $p$-groups of maximal nilpotency
class, given that the Sylow $p$-subgroups of the group $\rm{G}_{2}(p)$ for
$p\geq 5$ are of this type. Finally, Section 5 contains the proof of Theorems
A and B.

##  2\. preliminaries

We recall, following the approach of [DP15], the definition of orbit
categroies and Mackey functors adapted to the setting of fusion systems.

######  Definition 2.1.

Let $\mathcal{F}$ be a fusion system on a $p$-group $S$. The _orbit category_
of $\mathcal{F}$ is a category $\mathcal{O}(\mathcal{F})$ whose objects are
the subgroups of $S$ and whose morphisms are given by the left quotient:

$\mathrm{Hom}_{\mathcal{O}(\mathcal{F})}(P,Q)=\mathrm{Inn}(Q)\backslash\mathrm{Hom}_{\mathcal{F}}(P,Q).$

######  Remark 2.2.

Any morphism $\alpha:Q^{\prime}\to P$ in the orbit category is injective and
can be described as a composition of the equivalence class of an inclusion
with an isomorphism, i.e.

$\alpha=[\iota_{Q}^{P}][\varphi]$

where $\iota_{Q}^{P}$ is the inclusion map $Q\hookrightarrow P$ of a subgroup
$Q$ of $P$ and $\varphi:Q^{\prime}\to Q$ is an $\mathcal{F}$-isomorphism.

If $\mathcal{X}$ is an $\mathcal{F}$-overconjugacy closed collection of
subgroups of $S$ (that is, if $P\in\mathcal{X}$ then all overgroups of $P$ in
$S$ and all $\mathcal{F}$-conjugates of $P$ are contained in $\mathcal{X}$ as
well), then we have an analogous definition for $\mathcal{O}(\mathcal{X})$ in
which the obejcts are exactly the subgroups in $\mathcal{X}$. An important
example is given by the _centric-orbit category_
$\mathcal{O}(\mathcal{F}^{c})$ where $\mathcal{F}^{c}$ is the collection of
$\mathcal{F}$-centric subgroups of $S$, that is, all subgroups $P$ of $S$ such
that $C_{S}(Q)\leq Q$ for every subgroup $Q$ of $S$ that is
$\mathcal{F}$-isomorphic to $P$.

We will now recall the definition of _$\mathcal{X}$ -restricted Mackey functor
for $\mathcal{F}$_ as introduced in [DP15, Defintion 2.1], specializing it for
the finite field $\mathbb{F}_{p}$.

######  Definition 2.3.

Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$ and let
$\mathcal{X}$ be an $\mathcal{F}$-overconjugacy closed collection of subgroups
of $S$. A _$\mathcal{X}$ -restricted Mackey functor for $\mathcal{F}$_ is
given by a pair of functors

$M=(M^{*},M_{*}):\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$

satisfying the following conditions:

  1. (1)

(Bivariance) The functors $M^{*}$ and $M_{*}$ are respectively covariant and
contravariant and take the same values on all objects of
$\mathcal{O}(\mathcal{F})$. For all $P\leq S$ we denote $M^{*}(P)=M_{*}(P)$ by
$M(P)$.

  2. (2)

(Isomorphism) For all isomorphisms $\alpha$ in $\mathrm{Orb}(\mathcal{F})$ we
have $M^{*}(\alpha)=M_{*}(\alpha^{-1})$.

  3. (3)

($\mathcal{X}$-truncated Mackey decomposition) If $Q,R\leq P\leq S$ with
$Q,R,P\in\mathcal{X}$ we have

$M^{*}([\iota_{R}^{P}])\circ
M_{*}([\iota_{Q}^{P}])=\sum\limits_{x\in[R\backslash
P/Q]_{\mathcal{X}}}M_{*}([\iota_{R\cap\prescript{x}{}{Q}}^{R}])\circ
M^{*}([\iota_{R\cap\prescript{x}{}{Q}}^{\prescript{x}{}{Q}}])\circ
M_{*}([c_{x}|_{Q}]).$

where $[R\backslash P/Q]_{\mathcal{X}}$ denotes a set of representatives of
double cosets of $P$ of the form $RxQ$ with
$Q\cap\prescript{x}{}{R}\in\mathcal{X}$

When $\mathcal{X}$ consists of all subgroups of $S$, we simply say that $M$ is
a Mackey functor for $\mathcal{F}$.

If the functor $M$ is clear from the context we will adopt the following
notation: for subgroups $Q\leq P\leq S$ and an $\mathcal{F}$-isomorphism
$\varphi$ let

  * •

${\rm Ind}_{Q}^{P}:=M_{*}([\iota_{Q}^{P}]):M(Q)\to M(P)$;

  * •

${\rm Res}_{Q}^{P}:=M^{*}([\iota_{Q}^{P}]):M(P)\to M(Q)$;

  * •

$\mathrm{Iso}(\varphi):=M_{*}([\varphi])$.

######  Remark 2.4.

For any Mackey functor
$M:\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$,
the image $M(P)$ of $P\leq S$ always admits a structure of
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(P)$-module.

We will be interested in restricting the domain of a Mackey functor $M$ to the
centric-orbit category $\mathcal{O}(\mathcal{F}^{c})$. It is however not
always the case that such a restriction will satisfy the
$\mathcal{F}^{c}$-truncated Mackey formula.
In other words, using the notation of Definition 2.3, for a Mackey functor

$M=(M^{*},M_{*}):\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$

the restriction
$M|_{\mathcal{O}(\mathcal{F}^{c})}=(M^{*}|_{\mathcal{O}(\mathcal{F}^{c})},M_{*}|_{\mathcal{O}(F^{c})})$
is not necessarily an $\mathcal{F}^{c}$-restricted Mackey functor. A
sufficient condition is given by the following criterion.

######  Proposition 1.

[DP15, Proposition 4.4] Let $\mathcal{F}$ be a saturated fusion system on $S$
and $M$ a Mackey functor for $\mathcal{F}$ over a field of characteristic $p$.
Then the restriction $M|_{\mathcal{O}(\mathcal{F}^{c})}$ is an
$\mathcal{F}^{c}$-truncated Mackey functor for $\mathcal{F}$ if the composite

$M(P)\xrightarrow{{\rm Res}_{P\cap R}^{P}}M(P\cap R)\xrightarrow{{\rm
Ind}_{P\cap R}^{R}}M(R)$

is zero whenever $P,R\leq S$ are $\mathcal{F}$-centric and $P\cap R$ is not
$\mathcal{F}$-centric.

Given two Mackey functors
$M,N:\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
we say that $N$ is a subfunctor of $M$ if for all
$P\in\mathcal{O}(\mathcal{F})$ we have $N(P)\leq M(P)$. Accordingly, a Mackey
functor is _simple_ if it admits no subfunctor than itself and the zero
functor. The simple Mackey functors were characterized by Webb in [Web93] in
the case of finite groups and later adapted to the context of fusion systems
by Díaz and Park. We recall here their definition.

######  Definition 2.5.

Let $\mathcal{F}$ be a fusion system on a $p$-group $S$ and let $P,R\leq S$ be
such that $\mathrm{Iso}_{\mathcal{F}}(P,R)\neq\emptyset$. If
$\alpha\in\mathrm{Iso}_{\mathcal{F}}(P,R)$ and $V$ is a
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(P)$-module, then we denote by
$\prescript{\alpha}{}{V}$ the structure of
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(R)$-module on $V$ induced by the
following action

$[\gamma]\cdot v:=[\alpha^{-1}\gamma\alpha]v$

for all $[\gamma]\in\mathrm{Out}_{\mathcal{F}}(R)$ and $v\in V$.

######  Definition 2.6.

Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$. Consider a
subgroup $Q\leq S$ and a left
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(Q)$-module $V$. We are going to
define a Mackey functor
$S_{Q,V}:\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$.

  1. (a)

For any subgroup $L\leq S$ which is $\mathcal{F}$-conjugate to $Q$, fix an
$\mathcal{F}$-isomorphism $\alpha:Q\to L$ and define
$S_{Q,V}(L):=\prescript{\alpha}{}{V}$.

  2. (b)

For $P\leq S$ we define

$S_{Q,V}(P):=\bigoplus_{\alpha:Q\cong L\leq_{P}P}{\rm
tr}_{L}^{N_{P}(L)}(\prescript{\alpha}{}{V}),$

where the sum is indexed over the $P$-conjugacy classes of subgroups $L$ that
are $\mathcal{F}$-isomorphic to $Q$. We will sometimes denote the direct
summands of $S_{Q,V}(P)$ by $V_{\alpha,L}(P)$.

  3. (c)

Consider an isomorphism $\varphi:L_{1}\to L_{2}$ in
$\mathcal{O}(\mathcal{F})$. Suppose we fixed $\mathcal{F}$-isomorphisms
$\alpha_{i}:Q\to L_{i}$ for $i=1,2$. Then we define

$\displaystyle{\rm
iso}(\varphi):={S_{Q,V}}_{*}(\varphi):\prescript{\alpha_{1}}{}{V}$
$\displaystyle\to\prescript{\alpha_{2}}{}{V}$ $\displaystyle v$
$\displaystyle\mapsto[\alpha_{2}^{-1}\gamma\alpha_{1}]\cdot v,$

if $\varphi=[\gamma]$ for some
$\gamma\in\mathrm{Iso}_{\mathcal{F}}(L_{1},L_{2})$. Finally we define
${S_{Q,V}}^{*}(\varphi):={\rm iso}(\varphi^{-1})$.

  4. (d)

Consider an isomorphism $\varphi:P\to R$ in $\mathcal{O}(\mathcal{F})$. We
define ${\rm iso}(\varphi):={S_{Q,V}}_{*}(\varphi)$ on each summand
$V_{\alpha,L}(P)$ using point (c):

${\rm iso}(\varphi)|_{V_{\alpha,L}(P)}:={\rm iso}(\varphi|_{L}).$

Finally we define ${S_{Q,V}}^{*}(\varphi):={\rm iso}(\varphi^{-1})$.

  5. (e)

Consider an inclusion of subgroups $P\leq R$ of $S$ and the corresponding
morphism $\iota_{P}^{R}$ in $\mathcal{O}(\mathcal{F})$. We define ${\rm
Ind}_{P}^{R}:={S_{Q,V}}_{*}(\iota_{P}^{R}):S_{Q,V}(P)\to S_{Q,V}(R)$ on each
direct summand as follows:

$\displaystyle{\rm Ind}_{P}^{R}|_{V_{\alpha,L}(P)}:V_{\alpha,L}(P)$
$\displaystyle\to V_{\alpha,L}(R)$ $\displaystyle v$ $\displaystyle\mapsto{\rm
tr}_{N_{P}(L)}^{N_{R}(L)}(v).$

  6. (f)

The contravariant image, ${\rm
Res}_{P}^{R}:={S_{Q,V}}^{*}(\iota_{P}^{R}):S_{Q,V}(R)\to S_{Q,V}(P)$, is
defined as follows. Consider a direct summand $V_{\alpha,L}(R)$, if for all
$r\in R$ the conjugate $\prescript{r}{}{L}\nleq P$ then $V_{\alpha,L}(R)$ is
mapped to zero. Suppose otherwise that there are $R$-conjugates of $L$ in $P$,
fix the representatives for their $P$-conjugacy classes
$L_{i}=\prescript{r_{i}}{}{L}$ and the $\mathcal{F}$-isomorphisms
$\beta_{i}=c_{r_{i}}\circ\alpha$. Then $V_{\alpha,L}(R)$ is mapped to
$V_{\beta_{1},L_{1}}(P)\oplus\cdots\oplus V_{\beta_{k},L_{k}}(P)$ via the
diagonal map:

$v\mapsto v+\cdots+v.$

We now state and prove a property first uncovered by Webb [Web93, Proposition
2.8], in the language used in [DP15].

######  Lemma 2.7.

Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$ and let
$S_{Q,V}:\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
be the simple Mackey functor associated to the subgroup $Q\leq S$ and the left
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(Q)$-module $V$. Let $L\leq P\leq S$
be such that $L\in Q^{\mathcal{F}}$. If
$\alpha\in\mathrm{Iso}_{\mathcal{F}}(Q,L)$ and $V_{\alpha,L}(P)={\rm
tr}_{L}^{N_{P}(L)}(\prescript{\alpha}{}{V})\neq 0$, then $C_{P}(L)\leq L$.

###### Proof.

Suppose $C_{P}(L)\nleq L$, then $L<LC_{P}(L)\leq N_{P}(L)$ and

$V_{\alpha,L}(P)={\rm tr}_{L}^{N_{P}(L)}(\prescript{\alpha}{}{V})={\rm
tr}_{LC_{P}(L)}^{N_{P}(L)}{\rm tr}_{L}^{LC_{P}(L)}(\prescript{\alpha}{}{V}).$

The action of $N_{P}(L)$ on the
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(L)$-module $\prescript{\alpha}{}{V}$
is given by the composition of maps
$N_{P}(L)\to\mathrm{Aut}_{P}(L)\subseteq\mathrm{Aut}_{\mathcal{F}}(L)\to\mathrm{Out}_{\mathcal{F}}(L)$.
Both $L$ and $C_{P}(L)$ get mapped to the trivial subgroup under this
composition, hence their action on $\prescript{\alpha}{}{V}$ is trivial and so
${\rm tr}_{L}^{LC_{P}(L)}(\prescript{\alpha}{}{V})=0$ which implies
$V_{\alpha,L}(P)=0$, a contradiction. ∎

We now set a notation for the subgroups of $P$ that are
$\mathcal{F}$-conjugate to $Q$ and self-centralizing in $P$, whose importance
is highlighted by Lemma 2.7.

######  Definition 2.8.

Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$. For
$P,Q\leq S$, we define the set

$\textbf{cc}(P,Q)=\\{L\leq P\ |\
\mathrm{Iso}_{\mathcal{F}}(Q,L)\neq\emptyset{\rm\ and\ }C_{P}(L)\leq L\\}.$

######  Lemma 2.9.

Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$ and let
$S_{Q,V}:\mathcal{O}(\mathcal{F})\to\mathbb{F}_{p}{\operatorname{\mathbf{-mod}}}$
be the simple Mackey functor associated to the subgroup $Q\leq S$ and the left
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(Q)$-module $V$. Let $P\leq S$.

  1. (i)

If $S_{Q,V}(P)\neq 0$, then $\textbf{cc}(P,Q)\neq\emptyset$;

  2. (ii)

If $L\in\textbf{cc}(P,Q)$ and $x\in P$ then
$\prescript{x}{}{L}\in\textbf{cc}(P,Q)$.

  3. (iii)

If $L\in\textbf{cc}(P,Q)$ and $L\leq T\leq P$ then $L\in\textbf{cc}(T,Q)$.

###### Proof.

Suppose $S_{Q,V}(P)\neq 0$. By Definition 2.6(a)(b) there exists a subgroup
$L$ of $P$ such that $L\in Q^{\mathcal{F}}$ and
$V_{\alpha,L}(P)\neq\emptyset$. Part (i) now follows from Lemma 2.7.

Consider now $L\in\textbf{cc}(P,Q)$ and
$\alpha\in\mathrm{Iso}_{\mathcal{F}}(Q,L)$. For all $x\in P$ the composition
$c_{x}\circ\alpha:Q\to\prescript{x}{}{L}$ is still an isomorphism in
$\mathcal{F}$. Also, since $C_{P}(L)\leq L$ we have
$C_{P}(\prescript{x}{}{L})=\prescript{x}{}{C_{P}(L)}\leq\prescript{x}{}{L}$
proving $\prescript{x}{}{L}\in\textbf{cc}(P,Q)$. Thus part (ii) holds.

Finally, if $L\in\textbf{cc}(P,Q)$ and $L\leq T\leq P$, then $C_{T}(L)\leq
C_{P}(L)\leq L$ which implies $L\in\textbf{cc}(T,Q)$ and proves part (iii).

∎

##  3\. First results

In this section we work under the following

######  Hypothesis 3.1.

Suppose $p$ is an odd prime, $S$ is a $p$-group and $\mathcal{F}$ is a
saturated fusion system on $S$. Fix a subgroup $Q\leq S$ that is not
$\mathcal{F}$-centric and a simple
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(Q)$-module $V$.

######  Lemma 3.2.

Suppose Hypothesis 3.1 holds. If $K\leq S$ is abelian and $S_{Q,V}(K)\neq 0$,
then $K$ is $\mathcal{F}$-conjugate to $Q$. In particular $S_{Q,V}(K)=V$.

###### Proof.

By Lemma 2.9 the assumption $S_{Q,V}(K)\neq 0$ implies that there exists a
subgroup $L\leq K$ which is $\mathcal{F}$-conjugate to $Q$ and such that
$C_{K}(L)\leq L$. Since $K$ is abelian, we obtain $K=C_{K}(L)\leq L$ and so
$K=L$ is $\mathcal{F}$-conjugate to $Q$. ∎

From now on we consider the case of subgroups $Q,P$ and $R$ of $S$ for which
the composition of maps (1) is non-zero, that is, we assume

######  Hypothesis 3.3.

Suppose Hypothesis 3.1 holds and assume that $P$ and $R$ are
$\mathcal{F}$-centric subgroups of $S$ such that $T=P\cap R$ is not
$\mathcal{F}$-centric and the composition

(1) $S_{Q,V}(P)\xrightarrow{{\rm Res}_{T}^{P}}S_{Q,V}(T)\xrightarrow{{\rm
Ind}_{T}^{R}}S_{Q,V}(R)$

is non-zero.

It is clear that under the assumptions of Hypothesis 3.3 the group $T$ is
properly contained in both $P$ and $R$; in particular $P$ and $R$ are proper
subgroups of $S$.

######  Lemma 3.4.

Suppose Hypothesis 3.3 holds. Then

  1. (i)

$S_{Q,V}(P)$, $S_{Q,V}(R)$ and $S_{Q,V}(T)$ are all non-zero;

  2. (ii)

$T<P<S$ and $T<R<S$;

  3. (iii)

$Z(S)\leq T$; and

  4. (iv)

$P$ and $R$ are not abelian.

###### Proof.

Part (i) follows from the fact that the composition 1 is non-zero by
assumption. Part (ii) is a direct consequence of the fact that $P$ and $R$ are
$\mathcal{F}$-centric while $T$ is not $\mathcal{F}$-centric. Since $P$ and
$R$ are $\mathcal{F}$-centric, we also get $Z(S)\leq
C_{S}(P)\cap\mathrm{C}_{S}(R)\leq P\cap R=T$, proving (iii). Finally, aiming
for a contradiction, suppose $P$ is abelian. Then $P$ is
$\mathcal{F}$-conjugate to $Q$ by Lemma 3.2, contradicting the fact that $P$
is $\mathcal{F}$-centric while $Q$ is not $\mathcal{F}$-centric. The same
reasoning works for $R$. Therefore $P$ and $R$ are not abelian and this is
(iv). ∎

######  Lemma 3.5.

Suppose Hypothesis 3.3 holds. Then the following holds:

  1. (i)

$C_{S}(P)\leq T$;

  2. (ii)

$C_{S}(R)\leq T$.

###### Proof.

By Lemma 3.4(i) we get that $S_{Q,V}(P)$ and $S_{Q,V}(T)$ are nonzero. Hence
Lemma 2.9 (i) implies that both $\textbf{cc}(P,Q)$ and $\textbf{cc}(T,Q)$ are
nonempty. Aiming for a contradiction, suppose that for all
$L\in\textbf{cc}(P,Q)$ we have $L\nleq T$. By Lemma 2.7 and Definition 2.6(f)
we conclude that ${\rm Res}_{T}^{P}$ is the zero map, contradicting Hypothesis
3.3. So there exists $L\in\textbf{cc}(P,Q)$ such that $L\leq T$. Since $P$ is
$\mathcal{F}$-centric, we get

$C_{S}(P)=Z(P)\leq C_{P}(L)\leq L\leq T.$

So $C_{S}(P)\leq T$.
To prove part (ii) note that by Lemma 2.9 a necessary condition for ${\rm
Ind}_{T}^{R}$ to be nonzero is that there must exist $L\in\textbf{cc}(T,Q)$
such that $L\in\textbf{cc}(R,Q)$. For such subgroup $L$ we then have
$\mathrm{Z}(R)\leq C_{R}(L)\leq L\leq T$ and so $Z(R)\leq T$. But $R$ is
$\mathcal{F}$-centric, so we conclude $C_{S}(R)=Z(R)\leq T$ as desired. ∎

######  Lemma 3.6.

Suppose Hypothesis 3.3 holds and $T$ is abelian. Then

  1. (i)

$T$ is $\mathcal{F}$-conjugate to $Q$ and there exists
$\alpha\in\mathrm{Iso}_{\mathcal{F}}(Q,T)$ such that
$S_{Q,V}(T)=\prescript{\alpha}{}{V}$; and

  2. (ii)

if $T<C_{K}(T)\leq K\leq S$ then ${\rm tr}_{T}^{K}(S_{Q,V}(T))=0$.

###### Proof.

Since $T$ is abelian and $S_{Q,V}(T)\neq 0$ by Lemma 3.4, part (i) follows
from Lemma 3.2. In particular there exists
$\alpha\in\mathrm{Iso}_{\mathcal{F}}(Q,T)$ and
$S_{Q,V}(T)=\prescript{\alpha}{}{V}$.

If $T<C_{K}(T)\leq K$, then

${\rm tr}_{T}^{K}(\prescript{\alpha}{}{V})={\rm tr}_{C_{K}(T)}^{K}{\rm
tr}_{T}^{C_{K}(T)}(\prescript{\alpha}{}{V}).$

Since the action of $C_{K}(T)$ on $\prescript{\alpha}{}{V}$ (a
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(T)$-module) is trivial, by
definition of relative trace we obtain

${\rm tr}_{T}^{C_{K}(T)}(v)=[C_{K}(T):T]v,$

for all $v\in\prescript{\alpha}{}{V}$. As $C_{K}(T)$ and $T$ are $p$-groups
and we are working in characteristic $p$, the map ${\rm tr}_{T}^{C_{K}(T)}$ is
identically zero. Hence ${\rm tr}_{T}^{K}(\prescript{\alpha}{}{V})=0$, proving
(ii). ∎

######  Lemma 3.7.

Suppose Hypothesis 3.3 holds and $T$ is abelian.

  1. (i)

If $T$ is normal in $U\in\\{P,R\\}$ then $T=C_{U}(T)$.

  2. (ii)

If $T$ is normal in both $P$ and $R$ then
$\mathrm{Out}_{P}(T)\neq\mathrm{Out}_{R}(T)$.

###### Proof.

As $T$ is abelian, Lemma 3.6 (i) tells us that there exists an
$\mathcal{F}$-isomorphism $\alpha:Q\cong T$ and that
$S_{Q,V}(T)=\prescript{\alpha}{}{V}$. Let $U\in\\{P,R\\}$. As
$T\mathrel{\unlhd}U$, we have

$V_{\alpha,T}(U)={\rm tr}_{T}^{U}(\prescript{\alpha}{}{V}).$

Applying the definition of simple Mackey functors we see that if $U=P$ then
the restriction ${\rm Res}_{T}^{P}$ is defined as the zero map on those
summands $V_{\beta,L}(P)$ with $L\neq T$ and as the inclusion ${\rm
tr}_{T}^{P}(\prescript{\alpha}{}{V})\hookrightarrow\prescript{\alpha}{}{V}$
for $L=T$. So the first part of the composition of maps (1) reduces to ${\rm
tr}_{T}^{P}(\prescript{\alpha}{}{V})\hookrightarrow\prescript{\alpha}{}{V}$,
that is therefore non-zero by assumption. Similarly, if $U=R$ the induction
${\rm Ind}_{T}^{R}$ coincides with the relative trace map
$\prescript{\alpha}{}{V}\to{\rm tr}_{T}^{R}(\prescript{\alpha}{}{V})$, which
is again non-zero.

Now, if $T<C_{U}(T)$ for $U\in\\{P,R\\}$ then by Lemma 3.6 (ii) we have ${\rm
tr}_{T}^{U}(\prescript{\alpha}{}{V})=0$, contradicting what we just proved.
Hence (i) holds.

To prove (ii), suppose $\mathrm{Out}_{P}(T)=\mathrm{Out}_{R}(T)$. Note that,
as $T$ is abelian and normal in both $P$ and $R$, we have
$\mathrm{Out}_{P}(T)\cong P/T$ and $\mathrm{Out}_{R}(T)\cong R/T$. Moreover,
by what we showed above, the non-zero composition of maps (1) is reduced to

${\rm
tr}_{T}^{P}(\prescript{\alpha}{}{V})\hookrightarrow\prescript{\alpha}{}{V}\to{\rm
tr}_{T}^{R}(\prescript{\alpha}{}{V})$

Now the relative trace of an element $v\in\prescript{\alpha}{}{V}$ with
respect to the action of $R$ is

$\displaystyle{\rm tr}_{T}^{R}(v)$
$\displaystyle=\sum_{x\in[R/T]}[c_{x}|_{T}]\cdot v$
$\displaystyle=\sum_{c_{x}|_{T}\in\mathrm{Out}_{R}(T)}[c_{x}|_{T}]\cdot v$
$\displaystyle=\sum_{c_{y}|_{T}\in\mathrm{Out}_{P}(T)}[c_{y}|_{T}]\cdot v$
$\displaystyle=\sum_{y\in[P/T]}[c_{y}|_{T}]\cdot v$ $\displaystyle={\rm
tr}_{T}^{P}(v).$

Hence the maps $v\mapsto{\rm tr}_{T}^{R}({\rm tr}_{T}^{P}(v))$ and
$v\mapsto{\rm tr}_{T}^{P}({\rm tr}_{T}^{P}(v))$ coincide. This second map
however is the zero map as elements of ${\rm
tr}_{T}^{P}(\prescript{\alpha}{}{V})$ are invariant under the action of
$\mathrm{Out}_{T}(P)$, hence

${\rm tr}_{T}^{P}({\rm tr}_{T}^{P}(v))=\sum_{x\in[P/T]}[c_{x}|_{T}]\cdot{\rm
tr}_{T}^{P}(v)=\sum_{x\in[P/T]}{\rm tr}_{T}^{P}(v)=[P:T]{\rm tr}_{T}^{P}(v)=0$

where the last equality is justified by the fact that $[P:T]$ is a power of
$p$ and $\prescript{\alpha}{}{V}$ is an $\mathbb{F}_{p}$-vector space. This
proves $v\mapsto{\rm tr}_{T}^{R}({\rm tr}_{T}^{P}(v))$ is the zero map thus
composition (1) is also zero, a contradiction. ∎

######  Lemma 3.8.

Suppose Hypothesis 3.3 holds. Then $Z(S)<T$ and $T$ has order at least
$p^{2}$.

###### Proof.

Note that $Z(S)$ is an abelian normal subgroup of $S$ centralized by $S$.
Hence $Z(S)\neq T$ by Lemma 3.7(i) and we conclude using Lemma 3.4(iii). ∎

##  4\. The maximal class case

For $p\geq 5$, the Sylow $p$-subgroups of ${\rm G}_{2}(p)$ are $p$-groups of
maximal nilpotency class. For this reason, we now focus our attention on this
class of $p$-groups.

Suppose $S$ is a $p$-group of order $p^{n}$ having maximal nilpotency class.
Recall that the terms of the lower central series for $S$ are defined as

$\gamma_{2}(S):=[S,S];\quad\gamma_{i}(S):=[\gamma_{i-1},S]\quad\text{ for all
}3\leq i\leq n.$

Also, the terms of the upper central series of $S$ are

$\mathrm{Z}_{1}(S)=\mathrm{Z}(S);\quad\mathrm{Z}_{i}(S)/\mathrm{Z}_{i-1}(S)=\mathrm{Z}(S/Z_{i-1}(S))\quad\text{
for all }2\leq i\leq n-1.$

If $|S|\geq p^{4}$, we set

$\gamma_{1}(S):=\mathrm{C}_{S}(\gamma_{2}(S)/\gamma_{4}(S)$

and, following the terminology introduced in [GP22], we say that $S$ is
exceptional if $\gamma_{1}(S)\neq\mathrm{C}_{S}(\mathrm{Z}_{2}(S))$. Note that
both $\gamma_{1}(S)$ and $\mathrm{C}_{S}(\mathrm{Z}_{2}(S))$ are maximal
subgroups of $S$. We refer to [GP22] for a detailed analysis of the properties
of these groups.

If$p\geq 5$ and $S$ is a Sylow $p$-subgroups of ${\rm G}_{2}(p)$, then
$|S|=p^{6}$, $\gamma_{1}(S)$ is extraspecial and
$\mathrm{Z}(\gamma_{1}(S))=\mathrm{Z}(S)$. In particular $S$ is exceptional.

In this section, we assume the following:

######  Hypothesis 4.1.

Suppose Hypothesis 3.3 holds and $S$ is a $p$-group of maximal nilpotency
class.

A crucial role is played by the characterization of the essential subgroups of
saturated fusion systems on maximal calss $p$-groups:

######  Theorem 4.2.

[GP22, Theorem D] Suppose that $p$ is a prime, $S$ is a $p$-group of maximal
class and order at least $p^{4}$ and $\mathcal{F}$ is a saturated fusion
system on $S$. If $E$ is an $\mathcal{F}$-essential subgroup, then either
$|E|\leq p^{3}$, or $E=\gamma_{1}(S)$ or
$E=\mathrm{C}_{S}(\mathrm{Z}_{2}(S))$. Furthermore, if $S$ is exceptional,
then $|E|\neq p^{3}$.

Thanks to the work of Díaz and Park [DP15], we can exclude the case in which
the group $\gamma_{1}(S)$ is abelian:

######  Lemma 4.3.

Suppose Hypothesis 4.1 holds. Then $\gamma_{1}(S)$ is not abelian.

###### Proof.

The proof of [DP15, Theorem C] implies that the composition

$S_{Q,V}(P)\xrightarrow{{\rm Res}_{T}^{P}}S_{Q,V}(T)\xrightarrow{{\rm
Ind}_{T}^{R}}S_{Q,V}(R)$

is zero, contradicting our assumptions. ∎

######  Lemma 4.4.

Suppose Hypothesis 4.1 holds. If $T\mathrel{\unlhd}S$ then

  1. (i)

$T=\gamma_{i}(S)$ for some $i\geq 2$;

  2. (ii)

$\gamma_{i-1}(S)\leq C_{S}(T)$ is abelian (so in particular $T$ is abelian);
and

  3. (iii)

$C_{S}(T)$ is not a maximal subgroup of $S$.

In particular $T\neq Z_{2}(S)$ and $T\neq\gamma_{2}(S)$.

###### Proof.

Since $S$ has maximal nilpotency class and $T$ is normal in $S$ but not a
maximal subgroup of $S$ by Lemma 3.4 (ii), we deduce that $T=\gamma_{i}(S)$
for some $i\geq 2$. Note that $C_{S}(\gamma_{i}(S))=\gamma_{j}(S)$ for some
$j\geq 1$. Recalling that normal subgroups of $S$ that are $S$-centric are
also $\mathcal{F}$-centric, we conclude that $T$ must be properly contained in
its centralizer. Thus $\gamma_{i-1}(S)\leq C_{S}(T)$ and so $\gamma_{i-1}(S)$
is abelian, proving (ii). Now, aiming for a contradiction, suppose $C_{S}(T)$
is a maximal subgroup of $S$. Thus $|\mathrm{Aut}_{S}(T)|=[S\colon
C_{S}(T)]=p$. Since $T$ is abelian and normal in $S$, by Lemma 3.7 (i) we get
$T=C_{P}(T)=C_{R}(T)$. Thus $\mathrm{Aut}_{P}(T)\neq
1\neq\mathrm{Aut}_{R}(T)$, giving

$\mathrm{Aut}_{P}(T)=\mathrm{Aut}_{S}(T)=\mathrm{Aut}_{R}(T).$

This implies $\mathrm{Out}_{P}(T)=\mathrm{Out}_{R}(T)$ contradicting Lemma 3.7
(ii). Hence (iii) holds.

For the chaser of the Lemma, note that $C_{S}(Z_{2}(S))$ is a maximal subgroup
of $S$ (contrary to the statement of part (iii)) and if $T=\gamma_{2}(S)$ then
by part (ii) the group $\gamma_{1}(S)$ should be abelian, contradicting Lemma
4.3. ∎

######  Lemma 4.5.

Suppose Hypothesis 4.1 holds. Then either $T\leq\gamma_{1}(S)$ or $T\leq
C_{S}(Z_{2}(S))$.

###### Proof.

Aiming for a contradiction, suppose $T\nleq\gamma_{1}(S)$ and $T\nleq
C_{S}(Z_{2}(S))$. By Lemma 3.8 we have $Z(S)<T$ and $|T|\geq p^{2}$. So $P$
and $R$ are subgroups of $S$ of order at least $p^{3}$ not contained in
$\gamma_{1}(S)$. By [Gra18, Lemma 2.12] we deduce that $Z_{2}(S)\leq P\cap
R=T$. Note that $Z_{2}(S)\neq T$ by Lemma 4.4, so $|T|\geq p^{3}$. In
particular, Theorem 4.2 and the fact that $T$ is not $\mathcal{F}$-centric
imply that $T$ is not contained in any $\mathcal{F}$-essential subgroup of $S$
and so, by the Alperin-Goldschmidt’s fusion theorem ([AKO11, Theorem I.3.5]),
all $\mathcal{F}$-conjugates of $T$ can be found applying automorphisms of
$\mathrm{Aut}_{\mathcal{F}}(S)$.

Let $t\in T\setminus(\gamma_{1}(S)\cup\mathrm{C}_{S}(Z_{2}(S)))$. By [GP22,
Lemma 3.4] we have $C_{S}(t)=\langle Z(S),t\rangle$ and so by Lemma 3.4 (iii)
we get

$C_{S}(T)\leq C_{S}(t)=\langle Z(S),t\rangle\leq T.$

Thus $T$ is $S$-centric. Since $Z(S)$, $\gamma_{1}(S)$ and $C_{S}(Z_{2}(S))$
are characteristic subgroups of $S$ we deduce that all
$\mathcal{F}$-conjugates of $T$ contain $Z(S)$ and are not contained in
$\gamma_{1}(S)$ or $C_{S}(Z_{2}(S))$. The same reasoning used for $T$ then
shows that all its $\mathcal{F}$-conjugates are $S$-centric, impying that $T$
is $\mathcal{F}$-centric, a contradiction. This proves the statement. ∎

######  Lemma 4.6.

Suppose Hypothesis 4.1 holds and $\gamma_{1}(S)$ is extraspecial. Then the
following holds

  1. (1)

either $T\mathrel{\unlhd}S$ or $T\leq\gamma_{1}(S)=\mathrm{N}_{S}(T)$; and

  2. (2)

$T$ is not $S$-centric.

###### Proof.

By Lemmas 3.8 and 4.5 we have $Z(S)<T\subseteq\gamma_{1}(S)\cup
C_{S}(Z_{2}(S))$. Thus [GP22, Lemma 3.7] implies that either
$\mathrm{N}_{S}(T)\leq\gamma_{1}(S)$ or $T\mathrel{\unlhd}S$. If $T$ is not
normal in $S$, then $T\leq N_{S}(T)\leq\gamma_{1}(S)$ and since
$\gamma_{1}(S)$ is extraspecial and $[\gamma_{1}(S),\gamma_{1}(S)]=Z(S)\leq T$
we deduce that $\gamma_{1}(S)=\mathrm{N}_{S}(T)$.

As for the second statement, aiming for a contradiction, suppose $T$ is
$S$-centric, that is, $C_{S}(T)\leq T$. As $\mathcal{F}$ is saturated, if $T$
is fully normalized in $\mathcal{F}$ then it is $\mathcal{F}$-centric,
contradicting Hypothesis 4.1. Hence $T$ is $\mathcal{F}$-conjugate to a
subgroup $T^{\prime}$ of $S$ with $|N_{S}(T^{\prime})|>|N_{S}(T)|$. Using
Lemma 3.8 and the fact that $\gamma_{1}(S)$ is extraspecial, we get
$[\gamma_{1}(S),\gamma_{1}(S)]=Z(S)<T$ and so
$T\mathrel{\unlhd}\gamma_{1}(S)$. Thus $N_{S}(T)=\gamma_{1}(S)$ and
$T^{\prime}\mathrel{\unlhd}S$. Also, $|T|=|T^{\prime}|$, so $T^{\prime}$ is
not a maximal subgroup of $S$ and we deduce that
$T^{\prime}\leq\gamma_{1}(S)$. As $\gamma_{1}(S)$ is extraspecial and $T$ and
$T^{\prime}$ are normal subgroups of $\gamma_{1}(S)$ having the same order, we
deduce that

$|C_{S}(T^{\prime})|=|C_{S}(T)|.$

Therefore

$|C_{S}(T^{\prime})|=|C_{S}(T)|\leq|T|=|T^{\prime}|.$

Now, since $T^{\prime}$ is normal in $S$, which has maximal nilpotency class,
we have $T^{\prime}=\gamma_{i}(S)$ and $C_{S}(T^{\prime})=\gamma_{j}(S)$ for
some $i,j\geq 1$. Thus $C_{S}(T^{\prime})\leq T^{\prime}$. Now, $T^{\prime}$
is $S$-centric and fully normalized in $\mathcal{F}$, and so $T^{\prime}$ is
$\mathcal{F}$-centric. However, this implies that $T$ is $\mathcal{F}$-centric
as well, a contradiction. ∎

##  5\. Proof of Theorems A and B

We now focus on a Sylow $p$-subgroup of ${\rm G}_{2}(p)$, for $p\geq 5$,
obtaining the final ingredients to prove Theorems A and B.

###### Proof of Theorem B.

Aiming for a contradiction, suppose Theorem B is false. Hence there are
subgroups $Q,P,R\leq S$ and a simple
$\mathbb{F}_{p}\mathrm{Out}_{\mathcal{F}}(Q)$-module $V$ such that $P$ and $R$
are $\mathcal{F}$-centric, $Q$ and $T:=P\cap R$ are not $\mathcal{F}$-centric
and the composition

$S_{Q,V}(P)\xrightarrow{{\rm Res}_{T}^{P}}S_{Q,V}(T)\xrightarrow{{\rm
Ind}_{T}^{R}}S_{Q,V}(R)$

is non-zero. Hence Hypothesis 4.1 holds.

We first show that $\mathrm{N}_{S}(T)=\gamma_{1}(S)$. By Lemma 4.6 either
$\mathrm{N}_{S}(T)=\gamma_{1}(S)$ or $T\mathrel{\unlhd}S$. Aiming for a
contradiction, suppose the latter holds. By Lemma 4.4 we get $T=\gamma_{i}(S)$
for some $i\geq 2$ such that $\gamma_{i-1}(S)$ is an abelian subgroup of
$\gamma_{1}(S)$. Since $|S|=p^{6}$ and $\gamma_{1}(S)$ is extraspecial, we
deduce that either $T=Z(S)$ or $T=Z_{2}(S)$, contradicting Lemmas 3.8 and 4.4.
Hence $T$ is not normal in $S$ and we conclude that
$\mathrm{N}_{S}(T)=\gamma_{1}(S)$.

In particular $|T|\leq p^{4}$. Also, by Lemma 3.8 the group $T$ has order at
least $p^{2}$. If $|T|=p^{4}$ then $P$ and $R$ are maximal subgroups of $S$
and so $T=\gamma_{2}(S)$, contradicting the fact that $T$ is not normal in
$S$. Thus $p^{2}\leq|T|\leq p^{3}$. By Lemma 4.6, the group $T$ is not
$S$-centric, so if $|T|=p^{3}$ then it is not abelian.

Let’s now consider the groups $P$ and $R$. Recall that $P$ and $R$ are
distinct $\mathcal{F}$-centric subgroups of $S$ that are non abelian by Lemma
3.4 (iv). By Lemma 3.5 we have $Z(P)\leq T$ and $Z(R)\leq T$. Since
$Z_{2}(S)\neq T$ and if $|T|=p^{3}$ then $T$ is not abelian, we deduce that
$P,R\nleq C_{S}(Z_{2}(S))$.

Let $U\in\\{P,R\\}$.

Claim 1:

If $U\nleq\gamma_{1}(S)$ then $\gamma_{3}(S)\leq U\cap\gamma_{1}(S)<U$ and
$|U|\geq p^{4}$.

_Proof:_ Set $U_{0}=U\cap\gamma_{1}(S)$. By [Gra18, Lemma 2.12] we have
$U_{0}=\gamma_{i}(S)$ for some $i\geq 2$ and $[U\colon U_{0}]=p$. Since
$|U|\geq p^{3}$, we deduce that $Z_{2}(S)=\gamma_{4}(S)\leq U_{0}$, and so
$TZ_{2}(S)\leq U_{0}$. This proves $|U_{0}|\geq p^{3}$, and so
$\gamma_{3}(S)\leq U_{0}<U$ and $|U|\geq p^{4}$.

Claim 2:

If $U\leq\gamma_{1}(S)$ then either $U=\gamma_{1}(S)$ or $|U|=p^{4}$ and
$T=Z(U)$ has order $p^{2}$.

_Proof:_ Since $U$ is $S$-centric nonabelian, we deduce that either
$U=\gamma_{1}(S)$ or $|U|=p^{4}$ and $|Z(U)|=p^{2}$. In the latter case, since
$Z(U)\leq T$ and $|T|\leq p^{3}$, we deduce that $T$ is abelian and so
$|T|=p^{2}$, that is, $T=Z(U)$.

Claim 3:

If $U\leq\gamma_{1}(S)$ then $U=\gamma_{1}(S)$.

_Proof:_ Aiming for a contradiction, suppose $U<\gamma_{1}(S)$. Hence by Claim
2 $|U|=p^{4}$ and $T=Z(U)$. In particular, $T\mathrel{\unlhd}U$ and
$T<C_{U}(T)=U$, contradicting Lemma 3.7(i).

Suppose first that both $P$ and $R$ are not contained in $\gamma_{1}(S)$. Then
Claim 1 implies that $\gamma_{3}(S)\leq P\cap R=T$ and so $T=\gamma_{3}(S)$,
contradicting the fact that $T$ is not normal in $S$. Thus, without loss of
generality, we can suppose $P\leq\gamma_{1}(S)$. Hence by Claim 3 we have
$P=\gamma_{1}(S)$ and $R\nleq\gamma_{1}(S)$. Again by Claim 1,
$\gamma_{3}(S)<R$. therefore $\gamma_{3}(S)\leq P\cap R=T$, a contradiction.
This completes the proof. ∎

###### Proof of Theorem A.

By Theorem B, for any choice of the subgroups $Q,P$ and $R$ as in the
statement of Theorem B, the composition (1) is zero. Now Proposition 1
guarantees that the restriction $S_{Q,V}|_{\mathcal{O}(\mathcal{F}^{c})}$ is
an $\mathcal{F}^{c}$-truncated Mackey functor for $\mathcal{F}$. Hence [DP15,
Theorem A] implies that

${\lim\limits_{\longleftarrow}}_{\mathcal{O}(\mathcal{F}^{c})}^{i}{S_{Q,V}}^{*}|_{\mathcal{O}(\mathcal{F}^{c})}=0$

for all $i>0$. We conclude by [DP15, Proposition 4.3]. ∎

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