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Self-Dual Modules for Finite Groups of Odd Order

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Self-Dual Modules for Finite Groups of Odd Order CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Algebra 330 (2011) 418–430 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Self-dual modules for finite groups of odd order ∗ Ping Jin a,b, ,1 a School of Mathematical Sciences, Shanxi University, Taiyuan 030006, PR China b School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China article info abstract Article history: Let G be a finite group of odd order and let F be a finite field. Received 28 February 2010 Suppose that V is an FG-module which carries a G-invariant non- Available online 21 December 2010 degenerate bilinear form which is symmetric or symplectic. We Communicated by Michel Broué show that V contains a self-perpendicular submodule if and only if the characteristic polynomials of some specified elements of G Keywords: Self-dual module (regarded as linear transformations of V ) are precisely squares. Symplectic form This result can be applied to the study of monomial characters if Symmetric form the form on V is symplectic, and self-dual group codes if the form Hyperbolic module is symmetric. Self-dual group code © 2010 Elsevier Inc. All rights reserved. 1. Introduction Let G be a finite group and let F be a finite field. Suppose that V is a finite-dimensional FG- module which carries a G-invariant non-degenerate bilinear form b.Ifb is symmetric then we call V a symmetric F G-module, and if b is symplectic, that is, if b is skew-symmetric and in addition when F has characteristic 2, b(v, v) = 0forallv in V , then we call V a symplectic F G-module. Furthermore, given a symmetric or symplectic FG-module V , we say that V is hyperbolic if it ⊥ contains a self-perpendicular submodule W , namely, W is an FG-submodule satisfying W = W . Hyperbolic symplectic modules play an important role in the study of monomial characters and M-groups, see [1,7,9,10,14] and the references therein. Here we recall that an irreducible (complex) character of a finite group G is said to be monomial if it can be induced from a linear character of some subgroup of G, and the group G is called an M-group if all of its irreducible characters are monomial. * Address for correspondence: School of Mathematical Sciences, Shanxi University, Taiyuan 030006, PR China. E-mail address: [email protected]. 1 The author was supported by the NSF of Shanxi Province (No. 2008011011) and partially supported by the NSF of China (No. 10971054). 0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2010.12.003 P. Jin / Journal of Algebra 330 (2011) 418–430 419 On the other hand, hyperbolic symmetric modules naturally occur in self-dual (extended) group codes and self-dual permutation codes, see [2,15,16,18], for example. In coding theory, a group code is merely a left ideal of the group algebra FG and an extended group code is an FG-submodule of F ⊕ FG where G acts trivially on the additional component F .BecausebothFG and F ⊕ FG can be viewed as symmetric FG-modules with the form defined by the standard inner product, the self-dual codes are precisely the hyperbolic submodules. For permutation codes, the reader is referred to [2]. The main purpose of this paper is to find some necessary and sufficient conditions for a symmetric or symplectic FG-module V to be hyperbolic. An obvious necessary condition is that every self-dual simple FG-module occurs with even multiplicity in a composition series of V . This condition is also sufficient in many important cases. However, it is not so easy to test in practice. For finite groups of odd order, we will present an effective criterion for this condition by using characteristic polynomials of linear transformations, see Corollary B and Corollary C below. To be able to deal with symmetric FG-modules and symplectic FG-modules simultaneously, we ∗ will work with self-dual F G-modules. Recall that an FG-module V is said to be self-dual if V =∼ V ∗ as FG-modules, where V = HomF (V , F ) is the dual of V . It is well known that V is self-dual if and only if it carries a G-invariant non-degenerate bilinear form (see [5, VII.8.10]). Here we stress that a theme of the paper is modules over “small” fields, due to the following result (see [5, VII.8.22] or [14, Lemma 2.8]). Lemma 1. A group of odd order has no non-trivial absolutely irreducible self-dual modules over any field. To state our results we need to establish some notation that will be used freely in the paper. For any positive integers q and m with gcd(q,m) = 1, the least positive integer k satisfying the congruence k q ≡ 1 (mod m) is called the order of q modulo m,denotebyk = ordm(q).Wewriteπ(q) for the set of odd primes p with p q and ordp(q) even. For each integer i 1, we let πi (q) be the set of odd i = primes p such that p q and 2 is the highest 2-power of ordp(q).Clearlyπ(q) i1 πi (q). Moreover, for a set π of primes we call a positive integer n a π -number (or π -number)ifp ∈ π (or p ∈/ π ) for every prime divisor p of n.LetG be a finite group and g ∈ G. Then we say that g is a π -element if its order o(g) is a π -number, and that G is a π -group if its order |G| is a π -number. It is well known that g can be uniquely expressed as g = xy for some x, y ∈ G, such that x is a π -element, y is a π -element and xy = yx. Then we call x the π -part of g and y the π -part of g. When π consists of a single prime p,asusual,wealwaysreplaceπ and π by p and p , respectively. For the sake of simplicity, we say that a self-dual FG-module V is an even-multiplicity module, provided that every non-trivial self-dual simple FG-module occurs with even multiplicity in a com- position series of V . We note that Dade actually proved (in our words) that if V is a symplectic FG-module and G has odd order, then V is hyperbolic if and only if it is an even-multiplicity mod- ule. This easily follows by Proposition 2.1, along with Proposition 1.10 and Corollary 2.10 of [1] (see the proof of Corollary B below for details). In fact, this result also holds for symmetric modules under an extra assumption on the F -space V , see Corollary C below. Now the main result of the paper can be stated as follows. Theorem A. Let G be a group of odd order and let F be a finite field with q elements. For each i 1,let πi = πi (q) defined as above. If V = 0 is a self-dual F G-module, then the following conditions are equivalent. (1) V is an even-multiplicity F G-module. (2) For each i 1,therestrictionVC is an even-multiplicity F C-module for every cyclic πi -subgroup C of G. (3) For each i 1 and each πi -element g of G, the characteristic polynomial determined by g on V is of the form (x − 1)e f (x)2,wheree∈{0, 1} satisfying dim V ≡ e (mod 2) and f (x) ∈ F [x]. Note that in the situation of Theorem A, it easily follows that V is an even-multiplicity FG-module if and only if V C is also an even-multiplicity FC-module for every cyclic subgroup C. This “cyclic induction”, however, does not hold in general for groups of even order, as indicated by an example given in Section 3. In addition, we will present two examples in Section 2 to show that Theorem A is 420 P. Jin / Journal of Algebra 330 (2011) 418–430 no longer true if we replace the term “πi -element” in condition (3) by “p-element for every p ∈ πi ”, and that in Theorem A, not all of elements of G have the characteristic polynomial of the form given in (3) even if V is an even-multiplicity module. We will prove Theorem A in Section 3 by using some standard facts from characters and represen- tation theory. Actually, we will first reduce the group G in Theorem A to an M-group, and then to a cyclic group. This “cyclic case” will be treated alone in detail in Section 2. As an immediate application of Theorem A to symplectic FG-modules, we have the following criterion for the existence of self-perpendicular submodules. Corollary B. Let G be a group of odd order and let F be a finite field with |F |=q. If V = 0 is a symplectic F G-module, then V is hyperbolic if and only if for each i 1, the characteristic polynomial on V determined by every πi (q)-element of G is a square of a polynomial over F . It should be pointed out that Corollary B covers Loukaki’s theorem [14, Theorem A] that is re- markable but requires odd characteristic for the ground field. In fact, as a by-product we provide an independent proof of Loukaki’ theorem, neither restricting the characteristic of the ground field nor using Dade’s deep theorem [1, Theorem 3.2], although this paper and proof owe a lot to the approach of Dade and Loukaki.
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