Two-Groups in Which an Automorphism Inverts Precisely Half the Elements

2020

Discrete Mathematics

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR.First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR.Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.

Section: Factsmentioning

confidence: 99%

A conjecture on bipartite graphical regular representations

Yan

2020

Discrete Mathematics

“…Moreover,

N

has no automorphism inverting more than half of its elements. (Every group

N

that has an automorphism inverting half of its elements and no automorphism that inverts more is classified in by Hegarty and MacHale). (iv)

G

is isomorphic to

Q_{8}

, to

C_{3} \times C_{3}

or to

C_{3} \times C_{2}^{3}

. (v)

G

is generalised dihedral.…”

Section: Earlier Work and Preliminariesmentioning

confidence: 99%

“…We discuss in some detail the work in and establish some notation that we use for the rest of this section. Let

G

N

g

and

n_{0}

be as in Theorem (iii).…”

Section: The Groups In Theorem (Iii)mentioning

confidence: 99%

“…Proof By Theorem (iii),

G

is a 2‐group with a normal subgroup

N

| G : N | = 2

, and elements

g \in G ∖ N

and

n_{0} \in N

with

g^{2} = 1

; the action of

g

by conjugation on

N

inverts precisely half of the elements of

N;

and

N = H \cup n_{0} H

, where

H : = {n \in N ∣ n^{g} = n^{- 1}}

, and

| H | = | N | / 2

. From [, Lemma 1] and its proof (see also the last paragraph of [, p. 132]),

N

has an abelian subgroup

A

with

| N : A | = 4

and with

a^{g} = a^{- 1}

for each

a \in A

. Some more information on the interaction between

A

and

g

and

N

is available by reading the [, proof of Lemma 1] (again, see also the last paragraph of [, p. 132]).…”

Section: The Groups In Theorem (Iii)mentioning

confidence: 99%

“…From [, Lemma 1] and its proof (see also the last paragraph of [, p. 132]),

N

has an abelian subgroup

A

with

| N : A | = 4

and with

a^{g} = a^{- 1}

for each

a \in A

. Some more information on the interaction between

A

and

g

and

N

is available by reading the [, proof of Lemma 1] (again, see also the last paragraph of [, p. 132]). Let

1, x_{2}, x_{3}, x_{4}

be left coset representatives for

A

N;

thus

N = A \cup x_{2} A \cup x_{3} A \cup x_{4} A

.…”

Section: The Groups In Theorem (Iii)mentioning

confidence: 99%

See 2 more Smart Citations

Classification of finite groups that admit an oriented regular representation

Morris

Bull. London Math. Soc.

2018

This is the third, and last, of a series of papers dealing with oriented regular representations. Here we complete the classification of finite groups that admit an oriented regular representation (or ORR for short), and give a complete answer to a 1980 question of László Babai: 'Which [finite] groups admit an oriented graph as a DRR' ? It is easy to see and well understood that generalised dihedral groups do not admit ORRs. We prove that, with 11 small exceptions (having orders ranging from 8 to 64), every finite group that is not generalised dihedral has an ORR.

On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups

Yan

Journal of Graph Theory

2020

In this paper, we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let A be an abelian group and let ι be the automorphism of A defined by a a = ι −1 , for every a A ∈. A Cayley graph A S Cay(,) is said to have an automorphism group as small as possible if A S A ι Aut(Cay(,)) = , 〈 〉. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley digraphs. These results are used for the asymptotic enumeration of bipartite Cayley digraphs and graphs over abelian groups.