Prime Graphs of Finite Groups - Groups St Andrews · Prime Graphs of Finite Groups Hung P. Tong-Viet University of KwaZulu-Natal [email protected] Groups St Andrews 2013 University

Prime Graphs of Finite Groups

Hung P. Tong-Viet

University of [email protected]

Groups St Andrews 2013University of St Andrews

August 3-11, 2013

HP. Tong-Viet (UKZN) Prime Graphs of Finite Groups August 10, 2013 1 / 16

Table of contents

1 The Prime Graphs

2 Prime graphs without triangle

3 Sketch of Proofs


The Prime Graphs

Some Notation

Groups are finite.

Graphs are finite and simple (no loop or multiple edge).

Let G be a group.

Irr(G ) : the set of all complex irreducible characters of G .

cd(G ) = {χ(1) : χ ∈ Irr(G )} : character degree set of G .

ρ(G ) : the set of primes which divide some character degrees of G .

π(n) : the set of distinct prime divisors of n, with n a positive integer.

π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Some Notation

Groups are finite.


Let G be a group.





π(G ) := π(|G |).


The Prime Graphs

Prime graphs of groups

For a group G , there are two graphs associated to the set cd(G ).

The prime graph ∆(G ) :

vertex set: ρ(G )p 6= q ∈ ρ(G ) are joined to each other iff pq | a for some a ∈ cd(G ).

The degree graph Γ(G ) :

vertex set: cd(G ) \ {1}a 6= b ∈ cd(G ) \ {1} are joined to each other iff gcd(a, b) > 1.

The prime graph is used more often as it is compatible with normalsubgroups and factor groups.


The Prime Graphs




vertex set: ρ(G )

p 6= q ∈ ρ(G ) are joined to each other iff pq | a for some a ∈ cd(G ).





The Prime Graphs









The Prime Graphs









The Prime Graphs






vertex set: cd(G ) \ {1}

a 6= b ∈ cd(G ) \ {1} are joined to each other iff gcd(a, b) > 1.



The Prime Graphs









The Prime Graphs









The Prime Graphs









The Prime Graphs

Prime graphs of A5 and PSL2(8)

Figure : A disconnected graph with three connected components


The Prime Graphs

The prime graph of A8

2

7

5

3

Figure : Diamond


The Prime Graphs

The Main Questions

1 Which graph can occur as prime graphs of groups?

2 What is the structure of G if ∆(G ) is given?

In this talk, we are interested in the class of graphs which contain notriangle.

Examples of graphs without triangle

A cycle Cn with n ≥ 4

A tree Tn with n ≥ 1

A complete bipartite graph Km,n,m, n ≥ 1.

A path Pn with n ≥ 1 vertices.


The Prime Graphs

The Main Questions










The Prime Graphs

The Main Questions










The Prime Graphs

The Main Questions










The Prime Graphs

The Main Questions










The Prime Graphs

The Main Questions










The Prime Graphs

The Main Questions










The Prime Graphs

Known Results

(Zhang, 1998) If G is solvable, then ∆(G ) 6∼= P4.

(Lewis, Meng, 2012) If G is solvable and ∆(G ) ∼= C4 is a square, thenG = A× B where ρ(A) = {p, q} and ρ(B) = {r , s}.

(Lewis, White, 2013) If G is nonsolvable, then ∆(G ) is notisomorphic to P4 nor C4.

(Moreto, Tiep, 2008) ∆(G ) is not isomorphic to an octagon C8.

(Lewis, White, 2011) If Γ(G ) contains no triangles, then |cd(G )| ≤ 6.

Examples

Let G = 2 ·A6∼= SL2(9). Then cd(G ) = {1, 4, 5, 8, 9, 10} and Γ(G )

has a triangle but ∆(G ) has no triangle.

For G = PSL2(29), we have cd(G ) = {1, 15, 28, 29, 30}. Then ∆(G )has a triangle but Γ(G ) has none.


The Prime Graphs

Known Results


(Lewis, Meng, 2012) If G is solvable and ∆(G ) ∼= C4 is a square, thenG = A× B where ρ(A) = {p, q} and ρ(B) = {r , s}.(Lewis, White, 2013) If G is nonsolvable, then ∆(G ) is notisomorphic to P4 nor C4.



Examples





The Prime Graphs

Known Results





Examples





The Prime Graphs

Known Results





Examples





The Prime Graphs

Known Results





Examples





The Prime Graphs

Known Results





Examples





The Prime Graphs

Known Results





Examples





Prime graphs without triangle

Results

Theorem (H., 2013)

Let G be a group. Suppose that ∆(G ) has no triangle. Then

1 |ρ(G )| ≤ 5.

2 If |ρ(G )| = 5, then

(i) G ∼= PSL2(2f )× A, where |π(2f ± 1)| = 2 and A is abelian or(ii) G ∼= H × K , where H ∈ {A5,PSL2(8)}, K is solvable, ∆(K ) is

disconnected with two connected components, |ρ(K )| = 2 andρ(H) ∩ ρ(K ) = ∅.

Corollary (H., 2013)

Let G be a group. If ∆(G ) is a cycle or a tree, then |ρ(G )| ≤ 4.



Results

Theorem (H., 2013)


1 |ρ(G )| ≤ 5.2 If |ρ(G )| = 5, then

(i) G ∼= PSL2(2f )× A, where |π(2f ± 1)| = 2 and A is abelian or

(ii) G ∼= H × K , where H ∈ {A5,PSL2(8)}, K is solvable, ∆(K ) isdisconnected with two connected components, |ρ(K )| = 2 andρ(H) ∩ ρ(K ) = ∅.





Results

Theorem (H., 2013)


1 |ρ(G )| ≤ 5.2 If |ρ(G )| = 5, then







Results

Theorem (H., 2013)


1 |ρ(G )| ≤ 5.2 If |ρ(G )| = 5, then







Results

Theorem (H., 2013)


1 |ρ(G )| ≤ 5.2 If |ρ(G )| = 5, then







The first prime graph with 5 vertices

3 7

2

5 13

Figure : The prime graph of PSL2(26)



The second prime graph with 5 vertices

11

5

3

2

23

Figure : The prime graph of A5 × 231+2 : 11


Sketch of Proofs

Proofs

For solvable groups, using Pafly’s Condition, we obtain:

Lemma 1

If G is solvable and ∆(G ) has no triangle, then |ρ(G )| ≤ 4.

Thus, we can assume that G is nonsolvable.

Firstly, we look at almost simple groups.

Lemma 2 (Almost simple groups)

Let G be an almost simple group with simple socle S . Suppose that ∆(G )has no triangle. Then S ∼= PSL2(q) with q ≥ 4 a prime power,π(G ) = π(S) and |π(G )| ≤ 5.


Sketch of Proofs

Proofs


Lemma 1







Sketch of Proofs

Proofs


Lemma 1







Sketch of Proofs

Proofs


Lemma 1







Sketch of Proofs

Proofs


Lemma 1







Sketch of Proofs

Proofs, cont.

Next, we obtain a restriction on the structure of G

Lemma 3 (Reduction)

Let G be a nonsolvable group and let N be the solvable radical of G .Suppose that ∆(G ) has no triangle. Then there exists a normal subgroupM of G such that M/N ∼= PSL2(q), with q ≥ 4 a prime power, G/N is analmost simple group with socle M/N, and ρ(G ) = ρ(M).

The following lemma is crucial to the proof of our main theorem.

Lemma 4 (Key Lemma)

Let G be a nonsolvable group and let N be a normal subgroup of G suchthat G/N is nonabelian simple. Let θ ∈ Irr(N). Then either χ(1)/θ(1) isdivisible by two distinct primes in π(G/N) for some χ ∈ Irr(G |θ) or θextends to G and G/N ∼= A5 or PSL2(8).


Sketch of Proofs

Proofs, cont.


Lemma 3 (Reduction)



Lemma 4 (Key Lemma)



Sketch of Proofs

Proofs, cont.


Lemma 3 (Reduction)



Lemma 4 (Key Lemma)



Sketch of Proofs

Proofs, cont.


Lemma 3 (Reduction)



Lemma 4 (Key Lemma)



Sketch of Proofs

Proofs of the first part of the Theorem

Recall: G is a group and ∆(G ) has no triangle.

By Lemma 1, we can assume that G is nonsolvable.

By Lemma 2, G/N is almost simple with socle M/N andπ(G/N) = π(M/N).

Let τ = ρ(G )− π(G/N). Then τ ⊆ ρ(N).

By Lemma 4 and Pafly’s Condition, we deduce that |τ | ≤ 2.

If |π(G/N)| = |π(M/N)| ≤ 3, then |ρ(G )| = |π(G/N)|+ |τ | ≤ 5.

Assume |π(G/N)| ≥ 4. If τ 6= ∅, then there exists r ∈ τ andθ ∈ Irr(N) with r | θ(1). By Lemma 4, there exists χ ∈ Irr(G |θ) suchthat χ(1)/θ(1) is divisible by two distinct primes different from r .Hence ∆(G ) has a triangle, a contradiction. Thus τ is empty hence|ρ(G )| = |π(G/N)| ≤ 5 by applying Lemma 3.


Sketch of Proofs










Sketch of Proofs










Sketch of Proofs










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Sketch of Proofs

Possible extensions and related questions

Classify prime graphs of finite groups which is Kn-free for some n ≥ 4.

(Conjecture) If ∆(G ) is Kn-free with n ≥ 4, then |ρ(G )| ≤ 2n − 1.

(Strengthened Huppert’s ρ− σ Conjecture) ρ(G ) ≤ 2σ(G ) + 1.

Classify prime graphs of finite groups which contain a small numbert(G ) of triangles.

(Conjecture) For any group G , we have |ρ(G )| ≤ t(G ) + 5.


Sketch of Proofs

Thank you

Thank you for your attention!


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Prime Graphs of Finite Groups - Groups St Andrews · Prime Graphs of Finite Groups Hung P. Tong-Viet University of KwaZulu-Natal [email protected] Groups St Andrews 2013 University