Remarks on singular Cayley graphs and vanishing elements of simple groups

Let Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} be a finite graph and let A(Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\Gamma )$$\end{document} be its adjacency matrix. Then Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} is singular if A(Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\Gamma )$$\end{document} is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs Cay(G,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,H)$$\end{document} when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document} of G for which ∑h∈Hχ(h)=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{h\in H}\,\chi (h)=0.$$\end{document} At this stage, we focus on the case when H is a single conjugacy class hG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^G$$\end{document} of G; in this case, the above equality is equivalent to χ(h)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi (h)=0$$\end{document}. Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element h∈G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in G$$\end{document} is called vanishing if χ(h)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi (h)=0$$\end{document} for some irreducible character χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document} of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.

the eigenvalue 0. All graphs in this paper are undirected, without loops and without multiple edges; for all definitions please see Sect. 2.
Singular graphs play a significant role in graph theory, and there are many applications in physics and chemistry, see Sect. 2. While the literature on graph spectra is vast, it is not likely that a general theory of graph singularity per se will emerge. Some progress, however, can be made for graphs which admit a group of automorphisms that is transitive on the vertices of the graph. In some cases, the singularity problem then can be solved using techniques from ordinary character theory. The main purpose of this paper is to investigate these applications of character theory in graph theory.
In the following, G denotes a finite group and H denotes a connecting set in G. This is a subset of G such that (i) H does not contain the identity element 1 of G, (ii) H = H −1 := {h −1 | h ∈ H } and (iii) H generates G, that is, H does not lie in any proper subgroup of G. From these data, the Cayley graph = Cay(G, H ) with vertex set V = G and connecting sets H can be defined, see Sect. 2. Here is a regular graph of degree |H | and the group G acts transitively on the vertices of . Note though that a graph may be the Cayley graph of more than one group and connecting set.
In this paper, we specify the singularity problem to Cayley graphs Cay(G, H ) when the connecting set H is G-invariant, that is, H is a union of conjugacy classes of G. In this case, the following theorem (which is a special case of Theorem 1 in Zieschang [21]) reduces the singularity problem to a problem of character theory, see also the discussion in Sect. 2. In addition, χ(h) = 0 if and only if χ(h −1 ) = 0. Therefore, for constructing singular Cayley graphs it suffices to specify an irreducible character χ of G and a set X generating G such that χ takes the value 0 on X . Then, setting H = ∪ g∈G g(X ∪ X −1 )g −1 , we conclude that H is a connecting set and so Cay(G, H ) is singular.

Theorem 1.1 Let G be a group with a G-invariant connecting set H . Then Cay(G, H ) is singular if and only if there is an irreducible character
2. Burnside's theorem on character zeros [5, §32, Exercise 3] shows that every character χ of degree > 1 takes the value χ(h) = 0 for some h ∈ G. This implies that for every non-abelian group G there exists a singular Cayley graph on G, see Proposition 2.1.
3. If the character table of a group G is available explicitly (which is the case for all sporadic simple groups, say) then one can determine in principle all singular Cayley graphs Cay(G, M ∪ M −1 ) for G-invariant M.
In general, we have to look at elements g in G that take the value 0 for certain irreducible characters. Following [7], we say that g is non-vanishing if χ(g) = 0 for every irreducible character χ of G, otherwise we call g vanishing. Vanishing group elements are of particular interest in the block theory of finite groups. We postpone our comments on this matter until Sect. 4. Here we limit ourselves to the following well-known special case. Let |G| denote the order of G. If p is a prime then |G| p is the p-part of |G|, that is, |G p | is a power of p and |G|/|G| p is coprime to p. An element g in G is called p-singular if p divides the order of g.
The exceptions in this theorem are genuine. They can be detected easily for alternating groups A n with n = 7, 11, 13 by inspection of the character tables. However, for arbitrary n the problem of describing all non-vanishing elements in A n is still open. In any case, Theorem 1.5 yields many examples of singular Cayley graphs.
We first state some elementary results which yield a variety of singular Cayley graphs when G = A n . For g ∈ G let (c 1 , . . . , c k ), with c 1 ≥ c 2 ≥ · · · ≥ c k , be the cycle lengths of g, in the sense that g has k cycles where the longest cycle is of length c 1 , the second longest of length c 2 , and so on. Note that R 1 consists of all elements of G = A n fixing exactly one point of the natural G-set, whereas R 2 consists of elements fixing at most one point and having no 2-and 4-cycles in their cycle decomposition.
An element g ∈ G is called real if g −1 is conjugate to g. In symmetric groups, all elements are real. This is not the case for alternating groups. From Theorem 1.6, we deduce the following The proof of Theorem 1.6 is based on the Murnahgan-Nakayama formula for computing the values of irreducible characters of symmetric groups (see Sect. 4). In general, the problem of describing, for a given irreducible character χ of a given group G, the set {g ∈ G : χ(g) = 0} seems to be intractable, even when G is a symmetric or alternating group. However, the block theory of group characters supplies powerful tools for approaching this problem. In particular, we use block theory to prove Theorems 1.8 and 1.9.
Recall that for every prime p dividing the order of a finite group G the set of all irreducible characters of G is partitioned into blocks (or p-blocks to be accurate) and each block B determines a p-subgroup of G, defined up to conjugacy in G. This group is the defect group of B. If χ ∈ B then χ(g) = 0 whenever the p-part of g is not contained in a defect group of B. To use this fact, it is important to know the blocks with the smallest defect groups. If p > 3 then this smallest defect group is {1} for A n , see [8], and this yields Proposition 1.4. (Note that blocks with trivial defect group are called blocks of defect 0.) For G = A n and p = 2, the smallest defect groups can be easily determined (see Sect. 5), whereas for p = 3 this is still an open problem. This is discussed in [3, Theorem 2.1], where, for |G| 3 = 3 a and n = 7, the order of a smallest defect group is bounded from above by 3 (a−1)/2 . We improve this bound to 3 (a−1)/3 , see Proposition 6.5. Now we turn to the simplest (in a sense) version of the singularity problem for Cayley graphs: we assume that H = C ∪ C −1 where C = {1} is a single conjugacy class in G. Theorem 1.5 resolves this version of the singularity problem for Cayley graphs of the shape Cay(G, H ) with G simple, except when G = A n and when the elements of C have order 2 α 3 β for some α and β. Theorem 1.7 reduces the problem to the case where C = C −1 , that is, where H is a single conjugacy class. Below we state some partial results. One of them is the following: Theorem 1.8 Let G = S n or A n with n ≥ 5 and n = 7, 11, and let ω(G) be the set of element orders of G. Let ω 2,3 (G) be the set of all numbers in ω(G) that are not divisible by any prime p > 3. Then G contains a vanishing element of order m for every 1 = m ∈ ω 2,3 (G).
More can be said about the possible choices of h as an element of order m. By Theorem 1.5, there are no restrictions unless m is of the shape 2 α 3 β for some α and β. In the latter case, h can be chosen as any element of order m fixing a least number of members of the natural set {1, . . . , n}, see Theorem 6.8. Theorem 1.9 Let G = A n with n > 4 and let g ∈ G. Suppose that 2|g| and 3|g| are not in ω(G). Then g is vanishing unless n = 7. This statement is new only for |g| = 2 α 3 β , otherwise it follows from Lemma 1.2 and Proposition 1.4. It is not true that all elements satisfying the condition in Theorem 1.9 vanish at the same character of G. But if αβ = 0 then this is the case, see Corollary 3.2 and Proposition 6.4.
Notation Our notation for finite simple groups agrees with the Atlas [4]. In particular, A n means the alternating group on n letters, and S n is the symmetric group. The underlying set is often denoted by n , and it can be identified with {1, . . . , n}. For a set M ⊂ S n the support of M is supp(M) := {x ∈ n : gx = x for some g ∈ M}. In the other words, supp(M) is the complement in n of the set of the elements fixed by M.
If G is a group, then we write |G| for the order of G; if p is a prime then |G| p is the p-part of |G|, equivalently, the order of a Sylow p-subgroup of G. For nonzero integers m, n we denote the g.c.d. of m, n by (m, n). If g ∈ G then |g| is the order of g. The identity element of G is denoted by 1. For h ∈ G, we write h G for the conjugacy class of h in G. We write IrrG for the set of all irreducible characters of G. If χ is a character and M ⊂ G then χ(M) = 0 means that χ(g) = 0 for all g ∈ M. Let F be a field of characteristic 0. Then we denote by FV the vector space over F with basis V . This is a permutation module for the automorphism group of . The natural inner product on FV is given by

Singularity of graphs and Cayley graphs
The adjacency map α : FV → FV is the linear map given by . Therefore, α is symmetric with respect to this inner product. The matrix of α with respect to the basis V is the adjacency The singularity of graphs plays an important role in several parts of mathematics and applications. It would be impossible to review the vast literature on graph spectra in this paper in order to determine the occurrence of 0 as an eigenvalue. In representation theory and finite incidence geometry, the containment of one permutation character in another often is easiest to establish by showing that a certain graph is non-singular or that its nullity is bounded in a particular way. An example of this technique appears in proofs of the Livingstone-Wagner Theorem [14] about the representations of a permutation group G on the k-and (k + 1)-subsets of the set on which G acts.
We mention also the significance of graph singularity in systems analysis, physics and chemistry, see for instance the survey article [9]. Essentially, when modeling a discrete mechanical system (Hamiltonians), it is often necessary to work out a linear approximation of an operator where the constituents of the system and the relationships between them are represented by a finite graph. Many characteristics and observables of the system-its energy for instance-then typically involve the spectrum of this graph. This is one of the principles that underpins spectroscopy and Hückel Theory in chemistry [20]. In such applications, the singularity of a molecular graph of a feasible compound typically indicates that the compound is highly reactive, unstable, or nonexistent, see [9].
In order to investigate the singularity problem for graphs, we mention a few general facts so that our discussion here can be seen in a more general context. The problem becomes much more manageable for graphs which have a group of automorphisms that is transitive on the vertices of the graph. These includes in particular Cayley graphs for which we now give the basic definitions. It turns out that the singularity problem for vertex transitive graphs is equivalent to a nullity problem for Cayley graphs.
Let G denote a finite group with identity element 1. Then the subset H of G is a connecting set provided the following holds: Suppose now that H is a connecting set. Then define the graph = (V , E) with vertex set V = G by calling two vertices u, v ∈ G adjacent, denoted u ∼ v, if there is some h in H with hu = v. The first condition above is equivalent to saying that has no loops. The second conditions hold if and only if all edges are undirected, that is u ∼ v if and only if v ∼ u. The last condition is equivalent to saying that is connected. This graph is the Cayley graph = Cay(G, H ) on G with connecting set H . Its adjacency map α : FV → FV has the form for all vertices v in V = G. Since H = H −1 , the set of all neighbors of v ∈ is the set H v. In particular, is regular of degree |H |. Similarly, H Hv is the set of all vertices of distance ≤ 2 from v, and so on. The radius r ( ) of , as a graph invariant, is useful for studying generating sets in a group. Evidently, r = r ( ) is the least number r > 0 such that H r := {h 1 h 2 · · · h r | h i ∈ H } is equal to G, see property (iii). This invariant is a subject of intensive study by group theorists. For Cayley graphs, the eigenvalues of the adjacency map can be computed via the irreducible complex representations of the group, see Lovász [15], Babai [1], Zieschang [21], Brouwer and Haemers [2, Proposition 6.3.1], Diaconis and Shahsahani [6] and Ram Murty [18]. Theorem 1.1 is a part of Theorem 1 in [21] which states an explicit formula for eigenvalues when H is G-invariant. For the convenience of the reader, we provide a proof.
Proof of Theorem 1.1 Let ρ be the left regular representation of G on CG = CV , The adjacency map (2) above then becomes Let ρ i and α i denote the restriction of ρ and α to E i , respectively. Thus H ) is singular, say λ 1 = 0, then every irreducible representation ρ 1,i appearing in ρ 1 satisfies h∈H χ 1,i (h) = 0, where χ 1,i is the character of ρ 1,i . Conversely, if χ j,i is an irreducible character with h∈H χ j,i (h) = 0 then ρ j,i appears in ρ and so there is some E j on which λ j = 0.

Proposition 2.1 Let G be a non-trivial finite group whose order is not a prime. Then there exists a singular Cayley graph Cay(G, H ) for some connecting set H ⊂ G.
Proof: If G is a non-abelian simple group, then the result follows from Burnside's theorem on character zeros [5, §32, Exercise 3], saying that every nonlinear character (that is, of degree greater than 1) vanishes at some group element g ∈ G. Then we can take for H the union of the conjugacy classes of g and g −1 . If G has a proper normal subgroup K = 1 then the result follows from the following lemma (where we can take H = G \ K ):

Lemma 2.2 Suppose that H is a connecting set of G which is a union of cosets of K. Then the adjacency matrix of Cay(G, H ) is singular.
Proof The adjacency matrix of Cay(G, H ) has the form A = A * ⊗ J , where J is the (|K | × |K |)-matrix with all entries equal to 1, and A * is the adjacency matrix of the Cayley graph Cay(G/K , H /K ). As the rank of J is 1, we conclude that det A = 0, so Cay(G, H ) is singular.
Next we consider automorphisms of Cayley graphs. Let H be a connecting set in the group G and let = Cay(G, H ). Then for g ∈ G the right multiplication x → xg for x ∈ G is an automorphism of as is easy to see. Therefore, the right-regular representation of G on itself yields a regular group of automorphisms of . Cayley graphs are characterized by this property: [19]) The graph is isomorphic to a Cayley graph if and only if Aut( ) contains a subgroup that acts regularly on the vertices of .
By contrast, the left multiplication x → g −1 x for g and x ∈ G does not yield an automorphism of in general. It is easy to show that x → g −1 x is an automorphism of Cay(G, H ) if and only if g H = Hg. This is relevant for this paper as we are dealing with connecting sets that are unions of conjugacy classes. If H is a G-invariant connecting set then also left multiplication yields an automorphism of .
Finally we consider a connected graph = (V , E) which admits a vertex transitive group G of automorphisms. In this situation, Lovász [15] provides an associated Cayley graph * := Cay(G, H ) from which the spectrum of can be determined. For the convenience of the reader, we recall this construction here. Fix a vertex v ∈ V and let C be its stabilizer in G, with c := |C|. In view of Sabidussi's theorem, assume that Also, H generates G, this follows from the transitivity of G on vertices and the connectedness of . Therefore, we have a Cayley graph * : Comment: We see that the singularity problem for vertex transitive graphs can be reduced -in principle at least -to the nullity problem for Cayley graphs. The theorem can also be used to construct singular graphs: any graph with a vertex transitive but not vertex regular group of automorphisms yields a singular Cayley graph with the same group of automorphisms.

Elementary observations on zeros of alternating group characters
The comments in Sect. 1 suggest to pay particular attention to the alternating groups. In fact, the reasonings in this paper are mostly concerned with these groups. In this section, we collect a number of well-known facts about characters of alternating groups and prove some results on the zeros of some of their irreducible characters.
We first recall certain notions of the representation theory of S n . It is well known that the irreducible characters of S n are in bijection with the Young diagrams, and also with the partitions of n. So we write φ Y for the irreducible representation or the irreducible character of S n corresponding to a Young diagram Y . For a Young diagram Y , we write |Y | for the number of boxes in it. A subdiagram of Y is a Young diagram of S m for m < n which is contained in Y as a subset with the same top left hand corner. A box in Y is called extremal if there is no box either below or to the right of it. The set of all extremal boxes form the rim of Y .
The notion of a hook in a Young diagram Y is of common knowledge, see [12, page 55]. The number of boxes in a hook is called the length of it. A hook of length m is called an m-hook. The leg of a hook is the set of all boxes below the first row and ends in its foot. The number of the boxes in the leg is the leg length. The arm of the hook is its horizontal part, it ends in the hand of the hook, the right furthest box in the arm. Both foot and hand of the hook belong to the rim of Y .
To Below we need the Murnahgan-Nakayama formula [12, 2.4.7]. It expresses the character value of an irreducible character χ Y in combinatorial terms. Let g ∈ S n and g = ab, where a is an m-cycle and where b ∈ S n−m ⊂ S n is the permutation induced by g on the points fixed by a. The Murnahgan-Nakayama rule is the induction formula where the sum runs over all m-rims ν of Y and where i is the leg length of ν. (If no m-rim exists, then we have χ(g) = 0 by convention.) As an illustration, we state the following Lemma 3.1 Let G = A n or S n with n ≥ 7 and let M ⊂ G be the subset of all elements whose cycle decomposition has a cycle of length greater than 2 √ n + 2. Then Proof Let m be the minimal number i such that i 2 > n, so m > √ n. Let g ∈ M and let c(g) be maximal length of a cycle in the cycle decomposition of g. Then We view χ as a character of S n . Let g = g 1 b where g 1 is a cycle of size c 1 and the cycle lengths of b are (c 2 , . . . , c k ). By the Murnahgan-Nakayama rule (3) where ν runs over the c 1 -rims of Y and i is the leg length of ν. (If no c 1 -rim exists then χ(g) = 0.) Set r = c 1 . It is clear from the diagram shape that an r -rim is either a part of the first row (and then r ≤ n − 7) or Y 1 = Y \ ν is one of the diagrams [2, 2, 1], [2, 1 2 ] or [2]. In each case, there is exactly one way to delete ν, so the sum has at most one term.
Let n − r ≥ 7. Then Y 1 = [n − r , 3, 1] and the leg length of the r -rim ν is 0. So χ(g) = χ 1 (b). So we can use induction on k. The case with k = 1 follows from the above as then n − r = 0 < 7. If k > 1 then the cycle lengths of b are (c 2 , . . . , c k ), so the result follows by the induction assumption.

Proposition 3.4 Suppose that G is a doubly transitive permutation group on .
Then there is an irreducible character of G vanishing at every element g ∈ G fixing exactly one point of . In particular, if G = A n with n > 3 and if g fixes exactly one point then g vanishes at the irreducible character of degree n − 1.

Lemma 3.5 An element g ∈ A n is a real element if and only if one of the following conditions holds:
(1) The cycle decomposition of g has a cycle of even length.
(2) The cycle decomposition of g has two cycles of equal odd length. (Note that the fixed points are counted as cycles of length 1, so this includes any permutation that has two or more fixed points.) (3) All the cycles of g have distinct odd lengths c 1 , . . . , c k and k i=1 (c i − 1)/2 is even. In other words, the number of c i 's that are congruent 3 to modulo 4 is even.
Proof Clearly, g is conjugate to g −1 in S n . Note that the S n -conjugacy class of g is an A n -conjugacy class if and only if there is an odd permutation that centralizes g, these are the conditions 1) and 2). In the remaining case, a cycle of odd length 2 + 1 is inverted by an element of sign (−1) , and this gives the condition 3). Here the S n -conjugacy class of g splits in A n , but g is conjugate to g −1 in A n .
Proof of Theorem 1.7 If n ≥ 7, then any non-real element satisfies the assumption of Lemma 3.3, whence the result. If n ≤ 6 then either n = 6, |g| = 5 or n = 4, |g| = 3. In these cases, the result follows from Lemma 3.4. Finally, the claim that H is connected follows immediately if n > 4 as A n is simple. If n = 4 then H consists of all elements of order 3, so the claim follows by inspection of normal subgroups of A 4 .

Proposition 3.6 Let G be a simple group and suppose that g ∈ G is non-real. Then g is vanishing.
Proof By Theorems 1.5 and 1.7, we are left to check the sporadic groups. It is observed in [11, p. 414] that M 22 , M 24 are the only sporadic groups having non-identity nonvanishing elements, and these are of order 2, see the Atlas [4], and hence are real.

Blocks in symmetric groups and vanishing elements
In this section, we expose some part of representation theory of symmetric groups that is needed for the remainder of the paper. Recall the notation from Sect. 1. Let G be a finite group. For every prime p dividing |G| the irreducible characters of G partition into p-blocks. To every p-block, there corresponds a conjugacy class of p-subgroups of G and each of them is called a defect group of the block. If p d is the order of a defect group then d is called the defect of the block. In particular, blocks of defect 0 are those whose defect groups consist of one element. In addition, G has a p-block of defect 0 if and only if there is an irreducible character of G which has p-defect 0. See for instance Navarro [17] or Curtis and Reiner [5], Chapter VII, for general theory of blocks. We use the following well-known fact: Lemma 4.1 [10,Corollary 15.49] Let G be a finite group and let g ∈ G be a p-singular element. Let g = g p h = hg p , where g p , h ∈ g , g p is a p-element and |h| is coprime to p. Let χ be an irreducible character of G. Suppose g p is not contained in any defect group for the p-block containing χ . Then χ(g) = 0, in particular g is a vanishing element of G.
To use this lemma, one needs to know the defect groups of the p-blocks of S n (for p = 2 and 3). These are described in [12, Theorems 6.2.39 and 6.2.45] for any prime p. However, first we discuss a special case of blocks and characters of defect 0.

Characters of defect 0
For non-abelian simple groups, there is the following criterion for the existence of p-blocks of defect 0. (2): G has no 3-block of defect 0 if and only if G is isomorphic to Suz, C 3 , or A n with 3n + 1 = m 2 r , where r is square-free and divisible by some prime q ≡ 2 (mod 3).  (2) can be expressed in an alternative way. Specifically, A n has a 3-block of defect 0 if and only if n is of the form n = 3(x 2 1 + x 2 2 + x 1 x 2 ) + x 1 + 2x 2 , where x 1 , x 2 are integers, not necessary positive, see [13] or [8, p.333].

Remarks
(1) Applying Lemma 4.2 to a specific number n, and decomposing 3n + 1 as a product of primes, we obtain the following list of n < 60 with n ≡ 3 (mod 4) for which A n does not have a 3-block of defect 0: n = 13, 18, 28, 29, 38, 45, 46, 48, 53, 59. (2) It is not true that if G does not have an irreducible character of 3-defect 0 then there exists a 3-singular element g ∈ G such that χ(g) = 0 for every irreducible character χ of G, see the character table of Suz. Recall that every finite group G has a unique maximal normal nilpotent subgroup F(G), called the Fitting subgroup of G. Lemma 4.2 can be extended to non-simple groups as follows.

Proposition 4.5 [7, Theorem A] Let G be a finite group, and let h ∈ G be of order coprime to 6. Then either h belongs to the Fitting subgroup F(G) of G or h is vanishing.
For Cayley graphs, we therefore have the following general result: We mention the following recent result on zeros of characters in S n . This can be used to construct Cayley graphs of relatively high nullity.

Lemma 4.7 [16, Theorem 4.1]
Let p be a prime, let n ≥ p be a natural number and n = a 0 + a 1 p + · · · + a k p k its p-adic expansion. Let h be a p-element of S n whose cycle structure is 1 a 0 p a 1 ( p 2 ) a 2 . . . (p k ) a k . Let χ be an irreducible character of S n such that p divides χ (1). Then χ(h) = 0. This remains true for A n provided h ∈ A n . (See Remark following [16,Th 4.2]).

Blocks of symmetric groups
Let p be a prime. Every diagram that does not contain a p-hook is called a p-core. For instance, 2-cores are the diagrams of triangle shape [k, k − 1, . . . , 1]; in particular, S n has no 2-core of size n unless n = 1 + · · · + k = k(k + 1)/2 for some integer k > 0. Every diagram Y contains a unique p-core subdiagramỸ which is maximal subject to condition |Y | ≡ |Ỹ | (mod p). The key result of block theory of symmetric groups states that two irreducible characters are in the same block if their Young diagram yields the sameỸ [12, 6.1.21]. There is a simple algorithm to obtainỸ as follows.
If Y has no p-hook then Y =Ỹ . Otherwise remove arbitrary p-rim to obtain a subdiagram Y 1 . If Y 1 has a p-hook, remove some p-rim from Y 1 to obtain a subdiagram Y 2 and so on. The process stops if and only if one gets a subdiagramỸ which is a p-core. By [12,Theorems 2.7.16], this final subdiagramỸ is unique (independently from the p-hooks choice), and called the p-core of Y . Thus, |Ỹ | = |Y | − pb for some uniquely determined integer b ≥ 0, and this b is called the p-weight of Y (see [12, p. 80]). Note that the p-weight of a diagram is 0 if and only if the diagram is a p-core.
The following well-known fact follows easily from the dimension formula for irreducible characters in terms of hooks:

Lemma 4.8 Let χ be an irreducible character of S n labeled by a Young diagram Y. Then χ is of p-defect 0 if and only if Y is a p-core.
By [12, 6.1.35 and 6.1.42], there is a bijection between p-blocks of S n and the p-cores C such that |C| ≤ |n| and n − |C| ≡ 0 (mod p).

Theorem 4.9 [12, Theorems 6.2.39] Let χ be an irreducible character of S n labeled by the Young diagram Y , and let B be the p-block to which χ belongs. Let b be the p-weight of Y . Then a Sylow p-subgroup of S pb is a defect group of B.
Recall that the defect groups of a block are unique up to conjugacy. Here the group S pb is a natural subgroup of S n in the sense that this permutes pb elements of n , and fixes the remaining elements. (Note that if b = 0, then S pb is meant to be the identity group, and if n = pb then S pb = S n .) Moreover, the character of S n− pb corresponding toỸ is of defect 0.

Corollary 4.11 (1) Let B be a p-block of S n with defect group D and p-core C. Then D fixes exactly |C| elements of n .
It is easy to construct irreducible characters with given p-core C (provided n − |C| is a multiple of p): For our purpose, we are interested in the defect groups rather than in blocks themselves. Moreover, we can fix a defect group in every conjugacy class of defect groups in such a way that these defect groups form a chain with respect of inclusion. In fact, if the defect groups D, D are Sylow p-subgroups in S pb , S pb , resp., and b < b , then we can assume D ⊂ D . (For this, one can order the elements of and choose S pb to be the subgroup fixing elementwise the last n − pb elements of .) Therefore, with this ordering of defect groups it is meaningful to speak of the minimal defect group of S n , that is, the one with least possible b. Recall that the defect groups of a block are conjugate, and if D is one of them then the defect of a block is the number d such that |D| = p d . So a minimal defect group is a defect group of a block of minimal defect. Note that the maximal defect group is always a Sylow p-subgroup of S n . By Lemma 4.2, if p > 3 then the minimal defect group is trivial (that is, the group of one element). (Formally, the lemma is stated for A n but it remains true for S n .) If p > 2, then the defect groups of A n are exactly the same as those of S n ; if p = 2 then the defect groups of A n are of shape D ∩ A n for a defect group D of S n . Moreover, if χ is an irreducible character of S n reducible on A n then irreducible constituents belong to blocks whose defect groups are conjugate in S n , and hence have the same support. (This follows from [17, 9.26, 9.2].)

Lemma 4.13 Let B be a 2-block (resp., 3-block) of S n of nonzero defect. Then B contains an irreducible character that remains irreducible under A n .
Proof It is well known that the characters labeled by non-symmetric diagrams are irreducible under A n . So we show that B contains a character whose Young diagram is not symmetric.
Let Y = [l 1 , . . . , l k ] be the 2-core (resp. 3-core) diagram determined by B, and Y 1 the diagram obtained from Y by adding n − |Y | boxes to the first row of Y . As B is not of defect 0, n = |Y |, so n − |Y | = 2b (resp. 3b) for some integer b > 0. If l 1 ≥ k then Y 1 is not symmetric and the character labeled by Y 1 belongs to the block B (see Lemma 4.12). If l 1 < k (so p = 3) then take for Y 1 the diagram obtained from Y by adding 3b boxes to the first column, and conclude similarly. Lemma 4.14 Let G = S n or A n , g ∈ G, and let D 2 and D 3 be defect groups of a 2-block, or a 3-block, respectively. Suppose that |g| = 2 α 3 β and |supp(g)| > |supp(D 2 )| + |supp(D 3 )|. Then χ(g) = 0 for some irreducible character χ of G.
Proof Let G = S n and let g = g 2 g 3 = g 3 g 2 , where g 2 is a 2-element and g 3 is a 3-element of G. It is easy to observe that |supp(g)| ≤ |supp(g 2 )| + |supp(g 3 )|. So either |supp(g 2 )| > |supp(D 2 )| or |supp(g 3 )| > |supp(D 3 )|. In the former case, g 2 is not conjugate to an element of D 2 , so χ(g) = 0 for every irreducible character χ in a 2-block with defect group D 2 , by Lemma 4.1. Similarly, consider the latter case. Here the result follows for G = S n . If G = A n then the result follows from that for S n and the fact stated prior Lemma 4.13.
Comments In view of Lemma 4.14, it is desirable to determine the minimal defect group for every n and p = 2 or 3. If p = 2 then the number bp = 2b must be of shape k(k + 1)/2, so the minimal defect group of S n is a Sylow 2-subgroup of S 2b , where k is the maximal integer such that n − 2b = k(k + 1)/2 for some b > 0. If p = 3, we only prove the existence of a defect group D with |supp(D)| ≤ 2 √ n + 4 (Lemma 6.1), which is sufficient for our purpose.

Minimal defect group of S n for p = 2
The following lemma is obvious in view of the above comments. We first compute the minimal numbers in the set {|supp(D)| : D is a defect group of a 2-block of S n , 13 < n < 34}. Note that these are equal to n − t, where t is the maximal number of shape m(m + 1)/2 such that n − t is even.
Note that m(m + 1)/2 < 34 implies m ≤ 7 so t ∈ {1, 3, 6, 10, 15, 21, 28}. Suppose that a is odd. Then T m is not a 2-core of any diagram of S n . So consider T m−1 . Then n − |T m−1 | = a + m. If m is odd then a + m is even and then d = a + m ≤ 2m. Let m even. Then a + m is odd and a ≤ m − 1. It follows that T m−1 is not a 2-core of any diagram of S n . Consider T m−2 . Then n − T m−2 = a + 2m − 1 is even, so d ≤ 3m − 2.
Recall that for g ∈ S n and a prime p we denote by g p the element such that g = g p h, where h ∈ g and |h| is not a multiple of p. One observes that |supp(g p )| is the sum of cycle lengths divisible by p in the cycle decomposition of g. Table 1 The minimal defect group support for 2-blocks of S n for n < 34 Proof Let T be the triangle diagram of maximal size |T | ≤ n such that n − |T | is even.
Suppose first that |T | = n. Then, by Lemma 4.8, S n has a character χ of 2-defect 0. Recall that χ is a unique character in a block it is contained in. So for G = S n the statement follows from Lemma 1.2. If G = A n then χ | A n is the sum of two irreducible characters of degree χ(1)/2; moreover, each of them is of 2-defect 0 (indeed, an irreducible character χ of A n is of 2-defect 0 if and only if χ(1) is a multiple of |S n | 2 ; as |S n | 2 = 2|A n | 2 , it follows that χ(1)/2 is a multiple of |A n | 2 ). This implies the result for A n .
Let |T | < n and let Y be any diagram of size n containing T . Let χ be the irreducible character of S n labeled by Y . Then χ belongs to a block B, say, whose defect group D satisfies |supp(D)| = n −|T |. By Lemma 5.2, |supp(g 2 )| ≥ 3 √ 2n − 20 > n −|T |, so g 2 is not conjugate to an element of D. By Lemma 4.1, χ(g) = 0 for every irreducible character χ of B, whence the result for S n . It is known that the defect groups of blocks of A n to which the irreducible constituents of χ | A n belongs are D ∩ A n and conjugate in S n (see comments prior Lemma 4.13). So g 2 is not conjugate to an element of D ∩ A n , and the result follows as above for S n . By Lemma 4.13, Y can be chosen non-symmetric, so χ is irreducible under restriction to A n .
We say that the element g ∈ G ⊆ S n has maximal support if |supp(g)| ≥ |supp(h)| whenever h ∈ G and |h| = |g|. One easily observes that if g is of maximal support in G = A n and |g| is even then |supp(g)| ≥ n − 3. Proof Let g ∈ G be a 2-element of maximal support. Then |supp(g)| ≥ n − 3. If n > 13 then n − 3 ≥ 3 √ 2n − 20, and the result follows from Lemma 5.3. For n ≤ 13 and n = 5, 6, 8, 10, 12 the result follows as A n has a 2-block of defect 0. So we are left with n = 9, 13, which can be inspected by the character table of G in [4]. Let G = S n . Then we have to deal also with the cases with n = 5, 8, 12. If n = 12, the all elements of maximal support are in A 12 . The cases with n = 5, 8 follows by inspection.
Remark In A 7 , the elements of order 2 are not vanishing while all elements of order 4 are vanishing. In A 11 , the elements of order 2 and maximal support are non-vanishing while all other 2-elements are vanishing. In A 13 , the elements of maximal support and all 2-elements of order greater than 2 are vanishing, whereas all other 2-elements are non-vanishing (they form two conjugacy classes).
These diagrams appeared in [3, p. 1159]. One can show that these exhaust all 3-cores in S n .

Proof
We have to show that there is a 3-block with defect group D, say, such that |supp(D)| ≤ 2 √ n + 4. We can assume that S n has no block of defect 0 (otherwise D = 1 and the statement is trivial).

Remark
The bound 2 √ n + 4 in Lemma 6.1 is not sharp.
For g ∈ G = S n denote by g 3 the element such that g = g 3 h = hg 3 , where g 3 is a 3-element and |h| is coprime to 3.

Elements of maximal support whose order is divisible by 3
Let G = S n or A n and g ∈ G. Recall that g is of maximal support in G if |supp(g)| ≥ |supp(h)| for any element h of the same order as g. In other words, the number of fixed points of such an h does not exceed the number of fixed points of g.
Note that for g in G to be of maximal support depends on whether G = S n or A n . Say, if g ∈ A 6 is an involution (a double transposition) then g is of maximal support in A 6 but not in S 6 . The following is easily verified: Lemma 6.6 Let G = S n or A n and g ∈ G. Suppose that g is of maximal support.
Let G = A n . If χ | A n is irreducible then we are done. Otherwise, χ | A n is the sum of two irreducible characters of equal degree. Recall (Lemma 1.2) that an irreducible character χ of G os of p-defect 0 if and only if χ(1) ≡ 0 (mod |G| p ). Applying this to S n and A n , one observes that if χ is of p-defect 0 (with p ∈ {2, 3}) then so are the irreducible constituents of χ . So in this case the lemma follows from Lemma 1.2. Suppose that χ is not of defect 0. Then, by Lemma 4.13, this block χ belongs to has an irreducible character χ labeled by a non-symmetric Young diagram which is therefore irreducible under restriction to A n .
Obviously, Theorem 1.8 follows from the following result: Theorem 6.8 Let G = S n or A n with n > 4 and let g ∈ G. Suppose that g is of maximal support in G. Then χ(g) = 0 for some irreducible character of G, unless n = 7 or 11.
The cases with n < 13 can be examined from their character tables [4]. If 14 > n > 4 and n = 7, 11 then A n has a block of 3-defect 0. The non-vanishing elements g ∈ G = A 11 are in class 2B and 3A in notation of [4], and those in 3A are not of maximal support. Let G = A 7 . Then an element of maximal support is in class 3B or 6A, the latter is non-vanishing, see [4]. If n = 13 then the character table is available in the GAP library. One observes that the only non-vanishing elements in A 13 are in classes 2 A, 2B, 3A, and hence not of maximal support. This completes the proof.
If n = 17, 16, 14 then G has a 3-block of defect 0, and if n = 15, 12 then G has a 2-block of defect 0. If n ≤ 13 then the result follows by inspection of the character table of G.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.