Abstract
AbstractSuppose that $$\Delta $$
Δ
is a thick, locally finite and locally s-arc transitive G-graph with $$s \ge 4$$
s
≥
4
. For a vertex z in $$\Delta $$
Δ
, let $$G_z$$
G
z
be the stabilizer of z and $$G_z^{[1]}$$
G
z
[
1
]
the kernel of the action of $$G_z$$
G
z
on the neighbours of z. We say $$\Delta $$
Δ
is of pushing up type provided there exist a prime p and a 1-arc (x, y) such that $$C_{G_z}(O_p(G_z^{[1]})) \le O_p(G_z^{[1]})$$
C
G
z
(
O
p
(
G
z
[
1
]
)
)
≤
O
p
(
G
z
[
1
]
)
for $$z \in \{x,y\}$$
z
∈
{
x
,
y
}
and $$O_p(G_x^{[1]}) \le O_p(G_y^{[1]})$$
O
p
(
G
x
[
1
]
)
≤
O
p
(
G
y
[
1
]
)
. We show that if $$\Delta $$
Δ
is of pushing up type, then $$O_p(G_x^{[1]})$$
O
p
(
G
x
[
1
]
)
is elementary abelian and $$G_x/G_x^{[1]}\cong X$$
G
x
/
G
x
[
1
]
≅
X
with $$ \textrm{PSL}_2(p^a)\le X \le \mathrm{P\Gamma L}_2(p^a)$$
PSL
2
(
p
a
)
≤
X
≤
P
Γ
L
2
(
p
a
)
.
Funder
Università della Calabria
Publisher
Springer Science and Business Media LLC
Reference10 articles.
1. van Bon, J., Stellmacher, B.: Locally $$s$$-transitive graphs. J. Algebra 441, 243–293 (2015)
2. van Bon, J., Stellmacher, B.: On locally $$s$$-arc transitive graphs that are not of local characteristic $$p$$. J. Algebra 528, 1–37 (2019)
3. van Bon, J.: Four vertex stabilizer amalgams for locally $$s$$-arc transitive graphs of pushing up type. J. Algebra 633, 56–68 (2023)
4. van Bon, J.: On locally $$s$$-arc transitive graphs of pushing up type, in preparation
5. van Bon, J.: A family of locally $$5$$-arc transitive graphs of pushing up type, submitted