© Hindawi Publishing Corp. MODULAR REPRESENTATIONS OF LOEWY LENGTH TWO

Let G be a finite p-group, K a field of characteristic p ,a ndJ the radical of the group algebra K[G]. We study modular representations using some new results of the theory of extensions of modules. More precisely, we describe the K[G]-modules M such that J 2 M = 0 and give some properties and isomorphism invariants which allow us to compute the number of isomorphism classes of K[G]-modules M such that dimK (M) = µ(M) + 1, where µ(M) is the minimum number of generators of the K[G]-module M. We also compute the number of isomorphism classes of indecomposable K[G]-modules M such that dimK (Rad(M)) = 1.


Introduction.
Let K be a field of characteristic p > 0, and G a finite pgroup.The aim of this paper is to study K[G]-modules of Loewy length two; these are K[G]-modules M such that J 2 M = 0, where J is the radical of the group algebra K [G] (see [8]).Another characterization is the following: a K[G]module is of Loewy length two if and only if its annihilator I is not a maximal ideal containing J 2 .In general, the Loewy length of a K[G]-module M is the smallest integer n such that J n • M = 0; observe that n exists because K[G] is artinian, and J is a nilpotent ideal.The Loewy length is equal to the length of the kernel filtration of M (see [5]), defined by Modules of Loewy length n are described in Propositions 2.1 and 2.2; as a consequence, we find that every K[G]-module of Loewy length two is an extension of K s by K t , where s, t are positive integers, and G acts trivially on K s and K t .Since the Krull-Schmidt-Azumaya theorem holds for K[G]-modules, it is more convenient to study indecomposable K[G]-modules.In Proposition 3.1 and Corollary 3.2, we give necessary conditions for a K[G]-module of Loewy length two to be indecomposable.If M is indecomposable of Loewy length two, then J •M = M G , the subspace consisting of elements of M invariant under the G-action (see Theorem 3.4).In Proposition 3.7 we will see that this result is no longer valid if we consider modules of higher Loewy length.As another consequence of Theorem 3.4, we find that the minimum number of generators of an indecomposable K[G]-module M of Loewy length two, extension of K s by K t , is equal to s; in particular, if s = 1, then M is cyclic and dim K (M G ) = t.

Propositions 4.1 and 4.2 give a classification of indecomposable K[G]-modules
of Loewy length two.In Theorem 5.5, we determine the number η(s, m) of isomorphism classes of indecomposable K[G]-modules M of dimension s + 1 and of Loewy length two such that dim K (Rad(M)) = 1, and m = µ(G) is the minimum number of generators of G.We distinguish two different cases: if s > µ(G), then all modules are decomposable, and if s ≤ µ(G), then the number of isomorphism classes of indecomposable modules is equal to the number of s-dimensional subspaces of K m .Throughout this paper, K will be a commutative field of characteristic p > 0, G a finite p-group, and K[G] the group algebra.We denote by J the Jacobson radical of K[G] and µ(G) the minimum number of generators of G.A K[G]-module M will be assumed to be finitely generated; µ(M) is the minimal number of generators of M, and we will denote Also recall that the sequence [8, page 284]).The sequence (J k M) k is also called the J-filtration of M.

Preliminaries on extensions of modules.
Let Λ be a commutative ring and L and N two Λ-modules.We recall (see [1,6,10]) that an extension of L by N is a triple (f ,M,g), where M is a Λ-module and f , g are Λ-module homomorphisms such that the sequence is exact.Two extensions (f ,M,g) and (f ,M ,g ) of L by N are equivalent if there exists an isomorphism of Λ-modules φ : then it is well known (see [9], [3, Section 75], and [4, Proposition 25.10]) that where For a cocycle α ∈ Z 1 (G, T ), the corresponding extension is M = N × α L, which is equal to N × L as a vector space, with G-action g • (m, n) = gm + α(g)(gn), gn . (2.3) Proof.The module J n−1 M is a K[G]-module of Loewy length 1 and is isomorphic to K t , where t is a positive integer.The module M is an extension of Proof.The module M is an extension of M/JM by JM, and it is clear that Recall from [11] that We put Rg(α) = dim K (Im G (α)).Let {σ 1 ,σ 2 ,...,σ m } be a system of generators of G, where m = µ(G) is the minimal number of generators of G, and let (e 1 ,...,e s ) be a basis of L. We define α j (σ i ) = α(σ i )(e j ).It is then easy to see that {α j (σ i )|i ∈ {1,...,m}, j ∈ {1,...,s}} is a system of generators of Im G (α).Therefore, (2.7) So n ∈ Ker H (α), and therefore M H ⊆ N × α (Ker H (α)).The converse inclusion is easy.Proposition 3.7.Let G be a finite p-group, with p an odd prime and n an integer such that dim K (J n−2 /J n−1 ) ≥ 2. Then there exists an indecomposable Proof.The ring K[G] being an artinian and noetherian, J has a composition series.Hence, there exists an ideal and such that J n−2 /I is an irreducible K[G]-module (isomorphic to K).The ideal IJ is strictly included in J n−1 ; otherwise J n−2 is included in I, and therefore

Isomorphism classes of modules of Loewy length two.
When we study the modules of Loewy length two, we need to know when two modules M = K t × α K s and M = K t × β K s are isomorphic.This is what we do in the next proposition.
Proposition 4.1.Let M = K t × α K s and M = K t × β K s be indecomposable modules of Loewy length two.Then M and M are isomorphic if and only if there exist an automorphism ϕ of K t and an automorphism Proof.The proof is an adaption of the proof of [3,Lemma 81.8]: if M and M are indecomposable, then, by Theorem 3.4, M G = M G = K t × {0}, and therefore any homomorphism from M to M is a triangular matrix of homomorphisms and keeps K t globally invariant.For more details, the reader may consult [2].
Proof.If M and M are isomorphic, then there exist automorphisms ϕ and ψ of K t and K s such that ϕ The converse of Proposition 4.2 is not true in general; under a certain hypothesis, we have equivalence.
Proposition 4.3.Let G be a finite p-group and s, t two integers such that t = sµ(G).Let M = K t × α K s and M = K t × β K s be two indecomposable extensions of K s by K t .Then M M if and only if Rg(α) = Rg(β).
Proof.If M = K t × α K s and M = K t × β K s are indecomposable such that Rg(α) = Rg(β), then Rg(α) = Rg(β) = t = sm.The system (α(σ j )e i ) has t elements and generates K t , so it is a basis of K t .Similarly, (β(σ j )e i ) is a basis of K t .Then we define an automorphism ϕ of K t by ϕ(α(σ j )(e i )) = β(σ j )(e i ).The desired conclusion is deduced from Proposition 4.1.

Corollary 4.4. Let G be a finite p-group such that m = µ(G).
(1) There exists only one isomorphism class of cyclic K[G]-modules M of Loewy length two such that dim K (M) = m + 1. ( and all such modules are isomorphic to K m × α K. ( [8, page 119, Theorem 2.3]), and then I = J 2 .Remark 4.5.Proposition 4.3 asserts that if s and t are two integers such that t = sµ(G), then there exists only one isomorphism class of indecomposable K[G]-modules M of Loewy length two such that dim K (M G ) = t and µ(M) = s.

Extensions of K s by K.
We have seen in Corollary 3.6 that all indecomposable modules of a Loewy length two extension of K by K s are cyclic, and therefore they are isomorphic if and only if they have the same annihilator.In this section, we will study the isomorphism classes of modules of a Loewy length two extension of K s by K, with s being a nonzero integer.

Proposition 5.1. Let M = K × α K s be a K[G]-module extension of K s by K. Then M is an indecomposable K[G]-module if and only if Ker
Proof.If M = K × α K s and Ker G (α) = {0}, then we conclude from Proposition 2.4 that dim K (M G ) = 1, and therefore M is an indecomposable K[G]module.
From now on, we assume that K is the finite field of q = p r elements.From [7], we recall the following lemma.Lemma 5.2.Let V be a K-vector space of dimension n.The number of mdimensional subspaces of V is given by the formula Proof.Consider two homomorphisms f ,g : K m → K s .Then f and g are in the same orbit if and only if there exists an automorphism ϕ of K s such that ϕ • f = g.Now ϕ • f = g if and only if Ker f = Ker g, so two homomorphisms are in the same orbit if and only if they have the same kernel.Therefore, the number of orbits of the left action of Aut(K s ) on Hom K (K m ,K s ) is equal to the number of subspaces of K m of dimension k, where k is an integer such that (5.4) [8]).Since K s is a completely reducible K[G]-module, we have J • K s = 0, and therefore pr 2 (J • M) = 0. We have Propositions 2.4 and 3.1).Since pr 2 J • M = 0, we have -module of Loewy length one, which contradicts our hypothesis that M is of Loewy length two.We conclude that J •M = K ×{0}.
Let G be a finite abelian p-group with µ(G) = m (µ(G) is the minimal number of generators of G).Let η(s, m) be the number of isomorphism classes of K[G]modules of Loewy length two which are extensions of K s by K. Theorem 5.5.Let s be a positive integer and let G and m be as above.Then (5.5) Proof.Let T = Hom K (K s ,K) and take a basis {e 1 ,...,e s } of K s and a basis {e 1 ,...,e m } of K m .Also consider the map where -module of Loewy length two, then α ≠ 0 and Ker α = Ker G (α).
Consider a second K[G]-module M = K × β K s of Loewy length two and assume that ϕ : M → M is an isomorphism.It follows from Lemma 5.4 that [8]).Then there exists an automorphism ψ of K s such that for all i ∈ {1,...,m}, which is equivalent to so the number of isomorphism classes of K[G]-modules that are extensions of K s by K is equal to the number of orbits of the right action of Aut(K s ) on Hom K (K s ,K m )−{0}, which is equal to the number of orbits of the left action of Aut(K s ) on Hom K (K m ,K s ) − {0}, and our result follows from Proposition 5.3.
In order to study the quotient M G /J n−1 M, we need the following definition.(2) Let = {σ 1 ,σ 2 ,...,σ m } be a system of generators of G = (Z/pZ) m and consider an m × s matrix A = (a ij ) ∈ ᏹ m,s (K).Let M be the K-vector space with basis Ꮾ = {u 1 ,u 2 ,...,u s+1 }.We define the following G-action on M: (5.9) M together with the K[G]-module structure (5.9) will be denoted by K A .If s ≤ m and A is of rank s, then it can be shown that M = K A is an indecomposable K[G]-module of Loewy length two such that dim K (M) = s + 1 and µ(M) = s.More precisely, K A is isomorphic to K × α K s , where α ∈ Z 1 (G, T ), T = Hom K (K s ,K), and α(σ i )(e j ) = a ij .
(3) Every K[G]-module as in Corollary 5.8 is isomorphic to a certain K A .

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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. 1 ) 5 . 3 .
Proposition Let s and m be nonzero integers.The number of orbits of the left action of Aut(K s ) on Hom K (K m ,K s ) is equal to min(m,s)

Definition 5 . 6 .Corollary 5 . 7 . 3 . 5 . 8 . 7 . 5 . 9 .
Let M be a K[G]-module of Loewy length two.We call dim K (M G /JM) the index of M. By Theorem 3.4, the index of an indecomposable K[G]-module of Loewy length two is equal to zero.Let n be a nonzero integer and G a finite abelian p-group, with µ(G) = m.Then the number of isomorphism classes of K[G]-modules of Loewy length two, index r , minimal number of generators n and dimension n + 1 is equal to m (n−r ) .Proof.It follows from Proposition 2.4 and Theorem 3.4 that such a module is of the formK × α K n , with r = dim K (M G /J • M) = dim K (Ker G (α)) and Ker G (α) = Ker( α).Therefore, the number of isomorphism classes is equal to the number of orbits of the left action of Aut(K n ) on the set of homomorphisms f :K m → K n such that dim K (Im f ) = n −r (see Theorem 5.5), which is equal to m (n−r ) , by Proposition 5.Corollary Let G be a finite abelian p-group and s and m two positive integers.(1)If s ≤ m, then there exist m s isomorphism classes of indecomposable K[G]-modules which are extensions of K s by K.(2) If s > m, then all K[G]-modules which are extensions of K s by K are decomposable.Proof.If M is an indecomposable K[G]-module, then the index of M is equal to zero.The result is therefore an immediate consequence of Corollary 5.RemarkIf s = m, then there exists only one isomorphism class of an indecomposable K[G]-module which is an extension of K s by K. Indeed, m m = 1.This is a particular case of Proposition 4.3.Note 5.10.(1) Let M = N × α L be an R[G]-module extension of L by N and let L be a submodule of the R[G]-module L, and N a submodule of the R[G]module N contained in σ ∈G α(σ )(L ).Then we can consider α as a cocycle in Z 1 (G, T ), where T = Hom K (L ,N ), and N × α L is a submodule of the R[G]-module M.