The graph of the generating d-tuples of a finite soluble group and the swap conjecture

Crestani, Eleonora; Lucchini, Andrea

doi:10.1016/j.jalgebra.2012.11.026

Cited by 5 publications

(7 citation statements)

References 15 publications

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“…We can conclude that

a_{i} \in S

for each

i \in {1, \dots, τ}

. By [11, Cor. 5.2], there exist

r \leq diam true(Γ_{a, a}^{*} (S^{τ}) true)

and

c_{i} = (u_{i 1}, \dots, u_{i τ}) σ_{i} \in C

such that

false(ϕ, \dots, ϕ false) = c_{0} k c_{1} \dots k c_{r}

.…”

Section: Direct Powers Of Simple Groupsmentioning

confidence: 94%

“…The swap conjecture states that

{normalΣ}_{d} false(G false)

is connected for every finite group G and every

d \geq d (G)

. In [11] it was proved that this conjecture is true if

d > d (G)

, while in [16] it is proved that it is true also when

d = d (G)

and G is soluble. So we have: Corollary If G is a finite group and either

a + b > d (G)

a + b = d (G)

and G is soluble, then

{normalΓ}_{a, b}^{*} false(G false)

is connected.…”

Section: The Graphs Normalγabfalse(gfalse) and Normalγab∗false(gfalse)mentioning

confidence: 99%

“…In this section we will try to generalize some results proved in [11] concerning the generating graph of direct powers of non‐abelian simple groups. As a by‐product, we will see that the bounds on the diameter of the graphs

{normalΓ}_{a, b}^{*} false(G false)

, proved in Section 3, does not remain true if we drop the solubility assumption: for every positive integer η and every pair

a, b

of positive integers, a finite group G can be constructed such that

d (G) = a + b

and

{normalΓ}_{a, b}^{*} false(G false)

is connected with diameter at least η.…”

Section: Direct Powers Of Simple Groupsmentioning

confidence: 99%

“…Proof Clearly the star

{normalΓ}_{0, b}^{*} false(S^{τ} false)

is connected and

{normalΓ}_{1, 1}^{*} false(S^{τ} false)

is connected by [11, Thm. 3.1], so we may assume

0 < a \leq b

and

(a, b) \neq (1, 1)

.…”

Section: Direct Powers Of Simple Groupsmentioning

confidence: 99%

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Graphs encoding the generating properties of a finite group

Acciarri

Lucchini

2020

Mathematische Nachrichten

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Assume that G is a finite group. For every a,b∈N, we define a graph normalΓa,bfalse(Gfalse) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,⋯,xa) and (y1,⋯,yb) are adjacent if and only if ⟨x1,⋯,xa,y1,⋯,yb⟩=G. We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about G is encoded by these graphs.

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“…We can conclude that

a_{i} \in S

for each

i \in {1, \dots, τ}

. By [11, Cor. 5.2], there exist

r \leq diam true(Γ_{a, a}^{*} (S^{τ}) true)

and

c_{i} = (u_{i 1}, \dots, u_{i τ}) σ_{i} \in C

such that

false(ϕ, \dots, ϕ false) = c_{0} k c_{1} \dots k c_{r}

.…”

Section: Direct Powers Of Simple Groupsmentioning

confidence: 94%

“…The swap conjecture states that

{normalΣ}_{d} false(G false)

is connected for every finite group G and every

d \geq d (G)

. In [11] it was proved that this conjecture is true if

d > d (G)

, while in [16] it is proved that it is true also when

d = d (G)

and G is soluble. So we have: Corollary If G is a finite group and either

a + b > d (G)

a + b = d (G)

and G is soluble, then

{normalΓ}_{a, b}^{*} false(G false)

is connected.…”

Section: The Graphs Normalγabfalse(gfalse) and Normalγab∗false(gfalse)mentioning

confidence: 99%

{normalΓ}_{a, b}^{*} false(G false)

, proved in Section 3, does not remain true if we drop the solubility assumption: for every positive integer η and every pair

a, b

of positive integers, a finite group G can be constructed such that

d (G) = a + b

and

{normalΓ}_{a, b}^{*} false(G false)

is connected with diameter at least η.…”

Section: Direct Powers Of Simple Groupsmentioning

confidence: 99%

“…Proof Clearly the star

{normalΓ}_{0, b}^{*} false(S^{τ} false)

is connected and

{normalΓ}_{1, 1}^{*} false(S^{τ} false)

is connected by [11, Thm. 3.1], so we may assume

0 < a \leq b

and

(a, b) \neq (1, 1)

.…”

Section: Direct Powers Of Simple Groupsmentioning

confidence: 99%

See 2 more Smart Citations

Graphs encoding the generating properties of a finite group

Acciarri

Lucchini

2020

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

“…Tennant and Turner proposed the conjecture that Δ d (G) is connected (swap conjecture). In [4] it is proved that the free metabelian group of rank B Andrea Lucchini lucchini@math.unipd.it 1 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padua, Italy 3 does not satisfy this conjecture, but no counterexample is known in the class of finite groups. In [1] it was proved that the conjecture is true if d ≥ d(G) + 1.…”

Section: Introductionmentioning

confidence: 99%