PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 1, January 1998, Pages 65–66 S 0002-9939(98)04134-3

FUSION IN THE CHARACTER TABLE

GABRIEL NAVARRO

(Communicated by Ronald M. Solomon)

Abstract. Suppose that P is a Sylow p-subgroup of a finite p-solvable group G. If g ∈ P, then the number of G-conjugates of g in P can be read off from the character table of G.

One of the classical problems in character theory is to find what kind of group theoretical information the character table determines.

Theorem A. Let P be a Sylow p-subgroup of a finite p-solvable group G. If K is a conjugacy class of G, then |K ∩ P | can be read off from the character table of G.

Although, it is possible to find non-isomorphic non-p-solvable groups with the same character table, it does not seem easy to construct examples where the conclusion of Theorem A fails. In fact, we shall prove Theorem A by using some of the deep facts in Isaacs π-theory. We should say that it is not clear at all how to find a direct proof without it.

Proof of Theorem A. Let K be a conjugacy class of a π-separable group G and let H be a Hall π-subgroup of G. We prove that the integer |K ∩ H| can be read off from the character table of G.

By a theorem of G. Higman ((8.21) of [3]), we know that the character table determines the set of primes that divide the common order of the elements in K. Therefore, we may assume that K consists of π-elements (otherwise, |K∩H| is zero). By Higman’s Theorem, let K_{1}, …, K_{h} be the conjugacy classes of G consisting of π-elements, so that K is one of them. Write G^{0} = K_{1} ∪ … ∪ K_{h}, the set of πelements of G. Also, let us denote by α^{0} the restriction of a class function α of G to G^{0}. Among the functions χ^{0} for χ ∈ Irr(G), we consider the subset I_{π}(G) consisting of those functions that cannot be written in the form ν^{0} +μ^{0} for nonzero characters ν and μ of G. By Theorem A of [1], the set I_{π}(G) forms a basis for the vector space cf(G^{0}) of class functions on G^{0}. Hence, notice that, by using the character table, we may find {χ_{1}, …, χ_{h}} ⊆ Irr(G) such that I_{π}(G) = {(χ_{1})^{0}, …, (χ_{h})^{0}} (given χ ∈ Irr(G), since χ(1) is finite, there are only a finite number of characters ν, μ to check the equation χ^{0} = ν^{0} + μ^{0}). We may certainly assume that χ_{1} = 1_{G}.

Now, by Corollary (2.5) of [2], we have that 1_{H} is a ‘Fong character’ associated to (1_{G})^{0} and therefore,

[(χ_{i})_{H}, 1_{H}] = 0 for i ≥ 2 .
\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors August 5, 1996.

1991 Mathematics Subject Classification. Primary 20C15.

Research partially supported by DGICYT.

If x_{j} ∈ K_{j}, then

[(χ_{i})_{H}, 1_{H}] = \frac{1}{|H|} ∑_{j=1}^{h} |K_{j} ∩ H|χ_{i}(x_{j})

and thus, we may write

(|H| ; 0 ; ⋮ ; 0) = (\begin{matrix}χ_{1}(x_{1}) & … & χ_{1}(x_{h}) \\ ⋮ & ... & ⋮ \\ χ_{h}(x_{1}) & … & χ_{h}(x_{h}) \end{matrix}) (|K_{1} ∩ H| ; ⋮ ; |K_{h} ∩ H|) .

By Theorem A of [1], we know that the matrix

(\begin{matrix}χ_{1}(x_{1}) & … & χ_{1}(x_{h}) \\ ⋮ & ... & ⋮ \\ χ_{h}(x_{1}) & … & χ_{h}(x_{h}) \end{matrix})

is invertible. Since this matrix and |H| = |G|_{π} are clearly determined by the character table, we have that

(|K_{1} ∩ H| ; ⋮ ; |K_{h} ∩ H|)

is determined by the character table.

Although, in general, it is not clear how to calculate permutation characters from the character table, we may prove the following result.

Corollary B. Let P be a Sylow p-subgroup of a finite p-solvable group G. Then the permutation character (1_{P})^{G} can be read off from the character table of G.

Proof. Again, we assume that H is a Hall π-subgroup of a π-separable group G and we determine (1_{H})^{G} from the character table of G. If χ ∈ Irr(G), it is enough to show how to calculate [χ, (1_{H})^{G}] from the character table of G. If K_{1}, …, K_{h} are the conjugacy classes of G consisting of π-elements and x_{j} ∈ K_{j}, we have that

[χ, (1_{H})^{G}] = [χ_{H}, 1_{H}] = \frac{1}{|H|} ∑_{j=1}^{h} |K_{j} ∩ H|χ_{i}(x_{j}) .

The result now follows from Theorem A.

References

1. M. Isaacs, Characters of π-separable groups, J. Algebra, 86, (1984), 98-128. MR 85h:20012 2. M. Isaacs, Fong characters in π-separable groups, J. Algebra 99, (1986), 89-107. MR

87e:20015

3. M. Isaacs, Character Theory of Finite Groups, New York, Dover, 1994.

Departament d’Algebra, Facultat de Matematiques, Universitat de Valencia, 46100 Burjassot, Valencia, Spain

E-mail address: gabriel@uv.es