=Paper=
{{Paper
|id=Vol-1662/alg5
|storemode=property
|title=On the Gruenberg-Kegel graphs of finite groups
|pdfUrl=https://ceur-ws.org/Vol-1662/alg5.pdf
|volume=Vol-1662
|authors=Natalia V. Maslova
}}
==On the Gruenberg-Kegel graphs of finite groups==
https://ceur-ws.org/Vol-1662/alg5.pdf
On the Gruenberg–Kegel graphs of finite groups
Natalia V. Maslova
butterson@mail.ru
Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Russia)
Ural Federal University (Yekaterinburg, Russia)
Abstract
Let G be a finite group. The spectrum of G is the set ω(G) of orders of
all its elements. The subset of prime elements of ω(G) is called the prime
spectrum of G and is denoted by π(G). The spectrum ω(G) of a group G
defines its Grunberg–Kegel graph (or prime graph) Γ(G) with vertex set
π(G), in which any two different vertices r and s are adjacent if and only
if the number rs belongs to the set ω(G). We discuss some problems
concerning coincidence of Gruenberg–Kegel graphs of non-isomorphic
finite groups and of realizability of a graph as the Gruenberg–Kegel
graph of a finite group.
1 Terminology and Notation
During this paper by “group” we mean “a finite group” and by “graph” we mean “a finite undirected graph without
loops and multiple edges”. Our notation and terminology are mostly standard and can be found in [1, 3, 4, 7, 16].
Let G be a group. Denote by π(G) the set of all prime divisors of the order of G and by ω(G) the spectrum
of G, i.e. the set of all its element orders. The set ω(G) defines the Gruenberg–Kegel graph (or the prime graph)
Γ(G) of G; in this graph the vertex set is π(G) and different vertices p and q are adjacent if and only if pq ∈ ω(G).
Denote the number of connected components of Γ(G) by s(G) and the set of connected components of Γ(G) by
{πi (G) | 1 ≤ i ≤ s(G)}; for a group G of even order, we assume that 2 ∈ π1 (G). Denote by t(G) the greatest
cardinality of a coclique in the Gruenberg–Kegel graph of the group G and by t(r, G) the greatest cardinality of
a coclique in the Gruenberg–Kegel graph of the group G containing the prime r.
Let π be a set of primes. Denote by π 0 the set of the primes not in π. Given a natural n, denote by π(n) the
set of its prime divisors. Then π(|G|) is exactly π(G) for any group G. If |π(G)| = n then G is called n-primary.
A subgroup H of a group G is called a π-Hall subgroup if π(H) ⊆ π and π(|G : H|) ⊆ π 0 .
We will denote by S(G) the solvable radical of a group G (i. e. the largest solvable normal subgroup of G),
by F (G) the Fitting subgroup of G (i. e. the largest nilpotent normal subgroup of G), by Soc(G) the socle of G
(i. e. the subgroup of G generated by the set of all non-trivial minimal normal subgroups of G) and by Op (G)
the largest normal p-subgroup of G. A group G is almost simple if Soc(G) is a simple group. It’s well known, G
is almost simple if and only if there exists a simple group S such that S ∼ = Inn(S) E G ≤ Aut(S). In this case,
S∼ = Soc(G).
If G and H are groups, then we use notation G : H (G h H) for a split extension (semidirect product) of G
by (with, on) H.
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A.A. Makhnev, S.F. Pravdin (eds.): Proceedings of the 47th International Youth School-conference “Modern Problems in
Mathematics and its Applications”, Yekaterinburg, Russia, 02-Feb-2016, published at http://ceur-ws.org
26
Recall, a group G is a Frobenius group if there is a non-trivial subgroup C of G such that C ∩ gCg −1 = {1}
whenever g 6∈ C. A subgroup C is a Frobenius complement of G. Let
[
K = {1} ∪ (G \ gCg −1 ).
g∈G
Then K is a normal subgroup of Frobenius group G with a Frobenius complement C which is called Frobenius
core of G. Note, C and K are Hall subgroups of G and G = K h C.
Recall, that a n-clique (resp. a n-coclique) is a graph with n vertices in which all the vertices are pairwise
adjacent (resp. non-adjacent). A graph Γ is called bipartite with parts of sizes m and n if its vertices can be
divided into two non-empty disjoint subsets Vm and Vn such that |Vm | = m, |Vn | = n and the vertices from the
same subset are non-adjacent. We will denote by Km,n a complete bipartite graph whose vertices are adjacent if
and only if they belong to different subsets.
2 Finite groups and their Gruenberg-Kegel graphs
In the finite group theory many researchers are interested in various problems of the study of groups by their
arithmetical properties. One of such problems is the problem of the study of a group by some properties of its
Gruenberg–Kegel graph.
The Gruenberg–Kegel graph of a group is its fundamental arithmetical invariant, having numerous
applications.
General problem. Describe all groups whose Gruenberg–Kegel graphs have a given property.
One of the first results concerning prime graphs of groups was the following unpublished result of K. W.
Gruenberg and O. Kegel, which was published later in the paper of J. S. Williams [15, Theorem A].
Gruenberg–Kegel Theorem. If G is a group with disconnected Gruenberg–Kegel graph, then one of the
following statements holds:
(1) G is a Frobenius group;
(2) G is a 2-Frobenius group, i. e., G = ABC, where A and AB are normal subgroups of G, AB and BC are
Frobenius groups with cores A and B and complements B and C, respectively;
(3) G is an extension of a nilpotent π1 (G)-group by a group A, where S E A ≤ Aut(S), S is a simple
non-abelian group with s(G) ≤ s(S), and A/S is a π1 (G)-group.
Gruenberg–Kegel Theorem implies the complete description of solvable group with disconnected prime graphs:
they are exactly groups from items (1) and (2). Moreover, if G is a Frobenius group or 2-Frobenius group, then
Γ(G) has exactly two connected components (see [17]); if G is solvable, then each connected component of Γ(G) is
a clique. Williams and Kondrat’ev gave a description of connected components of the Gruenberg–Kegel graphs of
all simple non-abelian groups [15, 9]. In particular, they proved, if S is a simple non-abelian group then s(S) ≤ 6.
An adjacency criterion of the Gruenberg–Kegel graphs of simple groups and cocliques of maximal sizes in the
Gruenberg–Kegel graphs of simple groups are known (see [13, 14]).
In 1999 M. S. Lucido [10, Theorem 1] proved, if G is a group whose Gruenberg–Kegel graph contains a 3-
coclique, then G is non-solvable. But this result follows directly from earlier results by G. Higman and P. Hall.
Indeed, let G be a solvable group whose Gruenberg–Kegel graph Γ(G) contains a 3-coclique {p1 , p2 , p3 }. Using
Hall Theorem [3, Theorem 6.4.1] we conclude, that there exists a solvable {p1 , p2 , p3 }-Hall subgroup H of G,
such that its Gruenberg–Kegel graph Γ(H) is a 3-coclique. In view of Higman Theorem [8, Theorem 1] a solvable
group in which every element has prime power order is at most 2-primary. A contradiction.
A. V. Vasil’ev [12, Propositions 2, 3] proved a theorem which widely generalize Gruenberg–Kegel Theorem.
Theorem (A. V. Vasil’ev, 2005). Let us assume, that t(G) ≥ 3 and t(2, G) ≥ 2 for some group G. Then
the following conditions hold:
(1) there exists a simple non-abelian group S such that S E G/S(G) ≤ Aut(S);
(2) if ρ ⊆ π(G) is a coclique in Γ(G) with |ρ| ≥ 3, then at most one of the primes from ρ divides the product
|S(G)| · |G/S(G) : S|. In particular, t(S) ≥ t(G) − 1;
(3) one of the following conditions holds:
(a) every p ∈ π(G) which is non-adjacent to 2 in Γ(G) doesn’t divide the product |S(G)| · |G/S(G) : S|. In
particular, t(2, S) ≥ t(2, G);
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(b) there exists r ∈ π(S(G)) which is non-adjacent to 2 in Γ(G); in this case t(G) = 3, t(2, G) = 2 and S ∼
= A7
or P SL2 (q) for any odd q.
Moreover, Vasil’ev’s Theorem is valid for any non-solvable group G such that t(2, G) ≥ 2.
3 On the realizability of a graph as the Gruenberg-Kegel graph of a group
The Gruenberg–Kegel graph of a group G can be considered as a graph with |π(G)| vertices, where the vertices
are labeled by different primes and two different vertices p and q are adjacent if and only if there is an element
g ∈ G such that the order of g is pq. The following problem arises.
Problem 1. Let Γ be a graph with vertices labeled by different primes. Is there a group such that Γ is its
Gruenberg–Kegel graph?
Problem 1 has a negative solution in the general case. It can be shown by the following example.
Example 1 (see [2, § 3]). Let Γ be a 3-coclique with V (Γ) = {p, q, r}, where p, q and r are pairwise
distinct odd primes. Assume, G is a group such that Γ(G) = Γ. Then by Feit-Thompson theorem G is solvable
and by Gruenberg—Kegel theorem it is a Frobenius group or a 2-Frobenius group. Gruenberg—Kegel graphs
of Frobenius groups and of 2-Frobenius groups have exactly two connected components, but Γ(G) has three
connected components. This contradiction proves that there is no a group G such that Γ(G) = Γ.
We say that a graph Γ with |π(G)| vertices is realizable as the Gruenberg–Kegel graph of a group G if there
exists an one-to-one correspondence φ between the vertex set of Γ and π(G) such that the vertices x and y are
adjacent in Γ if and only if φ(x)φ(y) ∈ ω(G). In other words, there exists a vertices marking of Γ by different
primes from π(G) such that the marked graph is equal to Γ(G). A graph Γ is realizable as the Gruenberg–Kegel
graph of a group if Γ is realizable as the Gruenberg–Kegel graph of an appropriate group G. In other words, Γ
is isomorphic to the Gruenberg–Kegel graph of an appropriate group G.
As a generalization of Problem 1, we obtain the following
Problem 2. Let Γ be a graph. Is Γ realizable as the Gruenberg–Kegel graph of a group?
In view of Lucido’s result, it’s easy to see, 3-coclique is not realizable as the Gruenberg—Kegel graph of a
solvable group, but it is realizable as the Gruenberg—Kegel graph of the group A5 . However, Problem 2 has a
negative solution in the general case. It can be shown by the following example.
Example 2 (see [11, Lemma 5]). Let Γ be a n-coclique, where n ≥ 5. Then Γ is not realizable as the
Gruenberg–Kegel graph of a group. Suppose the converse. If Γ is realizable as the Gruenberg–Kegel graph of a
group G, then s(G) = n ≥ 5 and t(G) = n ≥ 5. In view of Lucido’s result, G is non-solvable. By Gruenberg–Kegel
Theorem there exists a simple non-abelian group S such that S E G/F (G) ≤ Aut(S) and s(S) ≥ s(G) = n.
According to [9, 15] we have n ≤ s(S) ≤ 6 and S is isomorphic either to J4 or to E8 (q) for q ≡ 0, 1, 4 (mod 5),
and consequently |π(S)| > 6. A contradiction.
There are just a few works devoted to problems of realizability of a graph as the Gruenberg–Kegel graph of
a group.
In 2008 I. N. Zharkov, who was a student of V. D. Mazurov, proved in his unpublished graduate work [18]
the following theorem.
Theorem (I. N. Zharkov, 2008). A chain is realizable as the Gruenberg–Kegel graph of a group if and
only if the length of this chain is at most 4.
In the paper [2] Problem 2 was solved for graphs with at most 5 vertices by A. L. Gavrilyuk, I. V. Khramtsov,
A. S. Kondrat’ev and the author. This investigation was initiated by A. Gavrilyuk in 2012 on the 43th
International Youth School-conference “Modern Problems in Mathematics and its Applications” and was
terminated by I. Khramtsov, A. Kondrat’ev and the author. The following theorem was proved.
Theorem (A. L. Gavrilyuk, I. V. Khramtsov, A. S. Kondrat’ev, N. M., 2014). Let Γ be a graph
with at most five vertices. Then
(1) If Γ is 5-coclique then Γ is not realizable as the Gruenberg-Kegel graph of a group.
(2) If Γ is not 5-coclique then Γ is realizable as the Gruenberg-Kegel graph of a group.
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4 On the realizability of graphs with 3-cocliques as the Gruenberg-Kegel graphs
of non-solvable groups
It’s interesting to obtain solutions of Problem 2 for some classes of graphs contain 3-cocliques. In [11] Problem
2 was solved for complete bipartite graphs, moreover for any complete bipartite graph which is realizable as
the Gruenberg–Kegel graph of a group it was obtained the “number” of realizations. The following theorem was
proved.
Theorem 1 (N.M., D. Pagon, 2016). Let Γ be a complete bipartite graph Km,n , where m ≤ n. Then the
following statements hold:
(1) Γ is realizable as the Gruenberg–Kegel graph of a group if and only if m + n ≤ 6 and (m, n) 6= (3, 3);
(2) if m + n ≤ 6 and (m, n) 6= (3, 3), (1, 5), then there exist infinitely many sets T of primes such that Γ is
realizable as the Gruenberg–Kegel graph of a group G and T = π(G);
(3) if (m, n) = (1, 5) and Γ is realizable as the Gruenberg–Kegel graph of a group G, then π(G) =
{2, 3, 7, 13, 19, 37}, O2 (G) 6= 1, and G/O2 (G) ∼
= 2 G2 (27).
2
3 7 13 19 37
The only marking of K1,5
5 Realizability of a graph as the Gruenberg-Kegel graph of a solvable group
In 2015 A. Gruber, T. Keller, M. Lewis, K. Naughton and B. Strasser [6] obtained a complete description of
graphs which are realizable as Gruenberg–Kegel graphs of solvable groups. They proved the following theorem.
Theorem (A. Gruber, T. Keller, M. Lewis, K. Naughton, B. Strasser, 2015). A graph Γ is realizable
as the Gruenberg–Kegel graph of a solvable group if and only if its complement is 3-colorable and triangle free.
Recently I. B. Gorshkov and the author investigated the question of isomorphism of Gruenberg–Kegel graphs
of solvable groups and almost simple groups. In [5, Theorem 1] the following theorem was proved.
Theorem 2 (I. B. Gorshkov, N.M., 2016). Let G be an almost simple group. Then the following conditions
are equivalent:
(1) Γ(G) does not contain a 3-coclique;
(2) Γ(G) is isomorphic to the Gruenberg–Kegel graph of a solvable group;
(3) Γ(G) is equal to the Gruenberg–Kegel graph of an appropriate solvable group.
The following question naturally arises.
Question. Is there a graph without 3-cocliques, whose complement is not 3-colorable, but which is isomorphic
to the Gruenberg–Kegel graph of an appropriate non-solvable group? In the other words, is there a graph which
is realizable as the Gruenberg–Kegel graph of an appropriate non-solvable group, but is not realizable as the
Gruenberg–Kegel graph of any solvable group?
Let q = pm and G be an almost simple group such that S = Soc(G) is isomorphic to one of the following
simple groups: An , where n ≥ 5; P SLn (q), where n ≥ 2 and (n, q) 6= (2, 2), (2, 3); P SUn (q), where n ≥ 3 and
(n, q) 6= (3, 2); P Spn (q), where n ≥ 4 is even; P Ωn (q), where n ≥ 7 is odd; P Ω±
n (q), where n ≥ 8 is even; an
exceptional group of Lie type over the field of order q; a sporadic simple group. Let f be the standard field
automorphism of S and g be the standard graph automorphism of S (see [4, 2.5.12, 2.5.13]).
In [19, Theorems 1, 3] M. R. Zinov’eva and V. D. Mazurov described simple groups whose Gruenberg–Kegel
graphs coincide with Gruenberg–Kegel graphs of Frobenius groups or 2-Frobenius groups. Using [13, 14] it is not
difficult to prove these simple groups are exactly all simple groups whose Gruenberg–Kegel graphs coincide with
Gruenberg–Kegel graphs of solvable groups. In [5, Theorem 2] the following theorem was proved.
Theorem 3 (I. Gorshkov, N.M., 2016). Let G be an almost simple group and S = Soc(G). Then Γ(G)
does not contain a 3-coclique if and only if π(G) = π(S) and one of the following conditions holds:
29
(1) S is isomorphic to one of the following groups: A9 , A10 , A12 , P SU3 (9), P SU4 (2), P Sp6 (2), P Ω+
8 (2),
3
D4 (2);
(2) G is isomorphic to one of the following groups: S5 , S6 , P GL2 (9), M10 , Aut(A6 ), S8 , Aut(P SL2 (8)),
Aut(P SL3 (2)), P GL3 (4)hf i, P GL3 (4)hgi, Aut(P SL3 (4)), P SL4 (4)hf i, P SL4 (4)hgi, Aut(P SL4 (4)),
Aut(P SU5 (2));
(3) G ∼= P GL2 (p), where p is either a Fermat prime or a Mersenne prime;
(4) S ∼= P SL2 (2m ), where m ≥ 4 is even and {2} ⊆ π(G/S);
(5) S ∼= P SL3 (p), where p is a Mersenne prime and (p − 1)3 6= 3;
(6) S ∼= P SL3 (p), where p is either a Fermat prime or a Mersenne prime, (p − 1)3 = 3 and Inndiag(S) ≤ G ≤
Aut(S);
(7) S ∼= P SL3 (2m ), where m ≥ 3, (2m − 1)3 = 3 and Inndiag(S)hgi ≤ G ≤ Aut(P SL3 (2m ));
(8) S ∼= P SL3 (2m ), where m ≥ 3, (2m − 1)3 6= 3 and Shgi ≤ G ≤ Aut(P SL3 (2m ));
(9) S ∼= P SL4 (2m ), where m ≥ 3 and Shgi ≤ G ≤ Aut(P SL4 (2m ));
(10) S ∼= P SU3 (p), where p is a Fermat prime and (p + 1)3 6= 3;
(11) S ∼= P SU3 (p), where p is a Fermat prime, (p + 1)3 = 3 and Inndiag(S) ≤ G ≤ Aut(S);
(12) S ∼= P SU3 (2m ), where m ≥ 2, (2m − 1)3 = 3, {2} ⊆ π(G/S) and Inndiag(S) ≤ G ≤ Aut(S);
(13) S ∼= P SU3 (2m ), where m ≥ 2, (2m − 1)3 6= 3 and {2} ⊆ π(G/S);
(14) S ∼= P SU4 (2m ), where m ≥ 2 and {2} ⊆ π(G/S);
(15) S ∼= P Sp4 (q).
Acknowledgements
The work is supported by Russian Science Foundation (project 14-11-00061): Theorem 1, by the grant of the
President of the Russian Federation for young scientists (grant no. MK-6118.2016.1), the Integrated Program for
Fundamental Research of the Ural Branch of the Russian Academy of Sciences (project no. 15-16-1-5) and the
Program of the State support of leading universities of the Russian Federation (agreement no. 02.A03.21.0006
of 27.08.2013): Theorems 2 and 3. The author is a winner of the competition of the Dmitry Zimin Foundation
“Dynasty” for support of young mathematicians in 2013 year.
References
[1] J. H. Conway, et al. Atlas of finite groups. Clarendon Press, Oxford, 1985.
[2] A. L.Gavrilyuk, I. V. Khramtsov, A. S. Kondrat’ev, N. V. Maslova. On realizability of a graph as the prime
graph of a finite group. Sib. Electron. Mat. Izv, 11:246–257, 2014.
[3] D. Gorenstein. Finite groups. Harper and Row, N. Y., 1968.
[4] D. Gorenstein, R. Lyons, R. Solomon. The classification of the finite simple groups. Number 3, Math. Surv.
Monogr., 40(3), RI: Amer. Math. Soc., Providence, 1998.
[5] I. B. Gorshkov, N. V. Maslova. Finite almost simple groups whose Gruenberg–Kegel graphs are equal to
the Gruenberg–Kegel graphs of solvable groups. To appear in Algebra and Logic.
[6] A. Gruber, et al. A Characterization of the prime graphs of solvable groups. J. Algebra, 442:397–422, 2015.
[7] F. Harrary. Graph theory. Addison-Wesley, Massachussets, 1969.
[8] G. Higman. Finite groups in which every element has prime power order. J. London Math. Soc. (2),
32:335–342, 1957.
[9] A. S. Kondrat’ev. Prime graph components of finite simple groups. Math. USSR Sb., 67:235–247, 1990.
[10] M. C. Lucido. The diameter of the prime graph of finite groups. J. Group Theory, 2(2):157–172, 1999.
[11] N. V. Maslova, D. Pagon. On the realizability of a graph as the Gruenberg-Kegel graph of a finite group.
Sib. Electron. Mat. Izv., 13:89–100, 2016.
30
[12] A. V. Vasil’ev. On connection between the structure of a finite group and the properties of its prime graph.
Sib. Math. J., 46(3):396–404, 2005.
[13] A. V. Vasil’ev, E. P. Vdovin. An Adjacency criterion for the prime graph of a finite simple group. Algebra
and Logic, 44(6):381–406, 2005.
[14] A. V. Vasil’ev, E. P. Vdovin. Cocliques of maximal size in the prime graph of a finite simple group. Algebra
and Logic, 50(4):291–322, 2011.
[15] J. S. Williams. Prime graph components of finite groups. J. Algebra, 69(2):487–513, 1981.
[16] R. Wilson, et al. Atlas of finite group representations. URL: http://brauer.maths.gmul.ac.uk/Atlas/
[17] H. Zassenhaus. Über endliche Fastkorper. Abhandl. math. Semin. Univ. Hamburg, 11:187–220, 1936.
[18] I. N. Zharkov. On groups whose prime graph is a chain. Bachelor work, Novosibirsk State University, 2008
(in Russian). = И. Н. Жарков. О группах, граф простых чисел которых является цепью. Выпускная
квалификационная работа бакалавра, Новосибирский государственный университет, 2008.
[19] M. R. Zinov’eva, V. D. Mazurov. On finite groups with disconnected prime graph. Proc. Steklov Inst. Math.,
283(S1):139–145, 2013.
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