In this paper, we investigate a suborbital graph for the normalizer of Γ0(N) ∈ PSL(2;R), where N will be of the form 24p2 such that p > 3 is a prime number. Then we give edge and circuit conditions ...on graphs arising from the non-transitive action of the normalizer.
A triangle group is denoted by
and has finite presentation
We examine a method for composition of permutation representations of a triangle group
that was used in Everitt's proof of Higman's 1968 ...conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations. In particular situations, where the representations being composed are all equivalent copies of an alternating group in the same degree containing at least one handle, then
has a quotient
. We also prove that if G has a quotient
containing at least two handles, then G has a quotient
for many
. This article contains the main results of the author's PhD thesis.
In this paper, we provide a method to determine when transitive imprimitive groups synchronize a transformation of non-uniform kernel, that is, we give the condition of
⟨
G
,
α
⟩
being a ...synchronizing semigroup where
G
is transitive imprimitive and
α
is transformation of non-uniform kernel.
A graph-theoretic environment is used to study the connection between imprimitivity and semiregularity, two concepts arising naturally in the context of permutation groups. Among other, it is shown ...that a connected arc-transitive graph admitting a nontrivial automorphism with two orbits of odd length, together with an imprimitivity block system consisting of blocks of size 2, orthogonal to these two orbits, is either the canonical double cover of an arc-transitive circulant or the wreath product of an arc-transitive circulant with the empty graph
K
¯
2
on two vertices.
For an arbitrary finite permutation group
G
, subgroup of the symmetric group
S
ℓ
, we determine the permutations involving only members of
G
as
ℓ
-patterns, i.e. avoiding all patterns in the set
S
ℓ
...\
G
. The set of all
n
-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals.
Classification of 2-arc-transitive dihedrants Du, Shaofei; Malnič, Aleksander; Marušič, Dragan
Journal of combinatorial theory. Series B,
11/2008, Volume:
98, Issue:
6
Journal Article
Peer reviewed
Open access
A complete classification of 2-arc-transitive
dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in D. Marušič, On ...2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162–196. The list consists of the following graphs:
(i)
cycles
C
2
n
,
n
⩾
3
;
(ii)
complete graphs
K
2
n
,
n
⩾
3
;
(iii)
complete bipartite graphs
K
n
,
n
,
n
⩾
3
;
(iv)
complete bipartite graphs minus a matching
K
n
,
n
−
n
K
2
,
n
⩾
3
;
(v)
incidence and nonincidence graphs
B
(
H
11
)
and
B
′
(
H
11
)
of the Hadamard design on 11 points;
(vi)
incidence and nonincidence graphs
B
(
PG
(
d
,
q
)
)
and
B
′
(
PG
(
d
,
q
)
)
, with
d
⩾
2
and
q a prime power, of projective spaces;
(vii)
and an infinite family of regular
Z
d
-covers
K
q
+
1
2
d
of
K
q
+
1
,
q
+
1
−
(
q
+
1
)
K
2
, where
q
⩾
3
is an odd prime power and
d is a divisor of
q
−
1
2
and
q
−
1
, respectively, depending on whether
q
≡
1
(
mod
4
)
or
q
≡
3
(
mod
4
)
, obtained by identifying the vertex set of the base graph with two copies of the projective line
PG
(
1
,
q
)
, where the missing matching consists of all pairs of the form
i
,
i
′
,
i
∈
PG
(
1
,
q
)
, and the edge
i
,
j
′
carries trivial voltage if
i
=
∞
or
j
=
∞
, and carries voltage
h
¯
∈
Z
d
, the residue class of
h
∈
Z
, if and only if
i
−
j
=
θ
h
, where
θ generates the multiplicative group
F
q
∗
of the Galois field
F
q
.
The additive groups of the residue ring ℤ
and of the vector space
over the field
(2), as well as the group
generated by these additive groups, share common imprimitivity systems and enter as ...subgroups into the Sylow 2-subgroup of the symmetric group
(ℤ
). These groups are used in cryptography as an encryption tool with the operations of addition in
and ℤ
. The permutation structure of the subgroups of the group
is presented. The kernels of homomorphisms which correspond to various systems of imprimitivity, the normal subgroups, and some modular representations of the group
over the field
(2) are described.
The distance from permutations of degree n = wr to the wreath product
with a fixed system of imprimitivity is investigated. In a number of cases, the value of the maximum distance is determined and ...the sets of the most distant permutations are described
On 2-arc-transitivity of Cayley graphs Marušič, Dragan
Journal of combinatorial theory. Series B,
2003, 2003-01-00, Volume:
87, Issue:
1
Journal Article
Peer reviewed
Open access
The classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83–86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: ...the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group
PGL(2,
q),
q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us—modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs—a step away from a complete classification of such graphs. As a byproduct, a partial description of 2-arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained.
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