We derive a family of prime ideals of the Burnside Tambara functor for a finite group G. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results ...towards the same result for a larger class of groups.
A linear configuration is said to be common in an Abelian group G if every 2-coloring of G yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked ...whether, analogously to a result by Jagger, Šťovíček and Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in Fpn for p≥5 and large n and in Zp for large primes p.
On infinite anticommutative groups Costantino Delizia; Chiara Nicotera
International journal of group theory,
03/2023, Volume:
12, Issue:
1
Journal Article
Open access
We completely describe the structure of locally (soluble-by-finite) groups in which all abelian subgroups are locally cyclic. Moreover, we prove that Engel groups with the above property are ...locally nilpotent.
In a finite group G,
ρ
(
G
)
denotes the product of element orders of G. Let
p
,
q
,
r
,
s
and t be prime numbers. Recently, it was proved that
C
p
3
, C
pq
,
C
p
2
q
and C
pqrs
are characterizable ...by the product of element orders, and if
ρ
(
G
)
=
ρ
(
C
pqr
)
, then
G
≅
A
5
or
C
pqr
. In this paper, we continue this work and we show that
C
p
×
A
5
(for p > 5) and C
pqrst
are characterizable by the product of element orders. In the rest of the paper, we show that if
ρ
(
G
)
=
ρ
(
C
pqr
)
s
ρ
(
C
s
)
p
2
q
,
then
(
p
,
q
,
r
,
s
)
=
(
2
,
3
,
5
,
11
)
and
G
≅
L
2
(
11
)
. This shows that the structure of
ρ
(
G
)
uniquely determined
L
2
(
11
)
and so
L
2
(
11
)
is the only group with this form of
ρ
(
G
)
.
We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism φ and use it to ...characterize when the homological stretch factor of φ is equal to its geometric stretch factor.
Let G be an abelian group and
be nonempty subsets of G. The sets
are said to form a complete decomposition of G of order k if
and
are pairwise disjoint. The size of a complete decomposition
of G is ...defined to be
In this paper, we determine the minimum and maximum size of a complete decomposition of a finite cyclic group.
•Automated detection method for cyclic symmetries is established.•Two theorems and corollaries for symmetry detection are given.•The proposed detection method is robust and applicable to both 2D and ...3D structures.•It gets unshifted nodes and members for different symmetry operations.•Low-order or 2D symmetric structures can be efficiently detected.
In general, for a complex engineering structure with a large number of nodes or members, the inherent symmetry is not easily recognizable. Even though someone succeeds in recognizing certain symmetry properties of the structure, these might be partial ones, and the others will be possibly unnoticed. To overcome this difficulty and enable the integration of computational analysis and symmetry methods, we propose an automated detection method for engineering structures with cyclic symmetries. Only the nodes and the connectivity patterns of the members are needed for implementing this algorithm. Using group theory, we first describe different cyclic groups of symmetries and their symmetry operations. In order to establish a group-theoretic algorithm for automated symmetry detection, several theorems and corollaries are presented. Then, on the basis of matrix representations of symmetry operations, the equivalence of the nodes and members of a structure is evaluated. Hence, the inherent symmetry operations of the structure are identified one by one. Illustrative examples show that the proposed automated symmetry detection method is robust and applicable to both 2D and 3D structures. Highly symmetric structures are recognized accurately and effectively. In addition, asymmetric structures and 2D structures can be recognized in a very small number of iterations.
The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend ...the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.
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