SEMIFIELDS ARISING FROM IRREDUCIBLE SEMILINEAR TRANSFORMATIONS
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1 J. Aust. Math. Soc. 85 (28), doi:.7/s SEMIFIELDS ARISING FROM IRREDUCIBLE SEMILINEAR TRANSFORMATIONS WILLIAM M. KANTOR and ROBERT A. LIEBLER (Received 4 February 28; accepted 7 September 28) Communicated by Martin W. Liebeck Dedicated to Cheryl Praeger for her sixtieth birthday Abstract A construction of finite semifield planes of order n using irreducible semilinear transformations on a finite vector space of size n is shown to produce fewer than n log 2 n different nondesarguesian planes. 2 Mathematics subject classification: primary 5E5; secondary 5A4, 7A35. Keywords and phrases: finite projective planes, semifield, semilinear transformation.. Introduction Let V = V K be a d-dimensional vector space over a finite field K. Suppose that T ƔL(V K ) is an irreducible semilinear transformation: and V are the only T -invariant subspaces of V. (The simplest example is V = K and T Aut(K ).) Then d T i K is a presemifield [5], so that there is a corresponding semifield plane π T (see Section 2 below). While it appears that there might be quite a few projective planes obtained in this manner, the purpose of this paper is to show that this is not the case. THEOREM.. Fewer than n log 2 n pairwise nonisomorphic nondesarguesian semifield planes π T of order n are obtained from irreducible semilinear transformations T on vector spaces of size n. A weaker bound announced in [6] highlighted remarks concerning the relatively small number of known semifield planes. Many standard results concerning linear transformations have been generalized to the semilinear case [4, 2], but these do not appear to give the desired information concerning irreducible transformations. In Section 3 we develop enough machinery concerning semilinear transformations to deduce the theorem. This research was supported in part by NSF grant DMS c 29 Australian Mathematical Society /9 $A
2 334 W. M. Kantor and R. A. Liebler [2] 2. Semifield planes A finite presemifield is a finite vector space V together with a product a b that is left and right distributive and satisfies the condition that a b = implies a = or b =. This produces an affine plane (a semifield plane [, Section 5.3]) with point set V 2 and lines x = c and y = m x + b. There is a simple, elegant construction of finite presemifields due to Jha and Johnson [5], using an irreducible semilinear transformation T on a d-dimensional vector space V over a finite field K. Namely, the set S T := d T i K consists of V additive maps V V, with all nonzero maps invertible; define a b = f (a)(b), a, b V, for an arbitrary additive isomorphism f : V S T. This produces a presemifield and hence also an affine plane π T. Different choices for f produce isomorphic planes π T [, p. 35]. We repeat the elementary proof in [6] that, if at least one of the k i K is not zero, then the element d T i k i of S T is invertible. If this transformation is not invertible then there is some nonzero vector v such that d T i (k i v) =. Then there is some j such that j d and T j (k j v) = j T i (k i v). Since T K = K T, we have T (K T j (v)) = K T (T j (k j v)) j K T i (v), so that the latter is a proper T -invariant subspace, whereas T is irreducible. If T is a linear transformation then this construction produces a field in the standard manner. In general, unlike in the case of fields, if T and T generate the same cyclic group then the planes π T and π T might not be isomorphic since S T is not T -invariant. However, ƔL(V )-conjugates of T produce ƔL(V )-conjugate sets S T and hence isomorphic planes π T (but not conversely, as is easily seen using GF( V )). Therefore, in the next section we focus on conjugacy of irreducible semilinear transformations. We begin with the following result. 3. Proof of Theorem. PROPOSITION 3.. Let T be an irreducible σ -semilinear transformation on a finite vector space V over a finite field K. Then there is a decomposition V = V V t (3.2) of V into subspaces V i permuted cyclically by T such that T t V is a -dimensional semilinear map over an extension field of K. Moreover, t divides the order of σ, and the map T t V uniquely determines T up to GL(V )-conjugacy. PROOF. We will proceed in several steps. Throughout the proof, V will always denote a vector space over K. Whenever a subspace of V is viewed as a vector space over another field, or the field involved needs to be emphasized, we will add that field as a subscript. STEP. Let s be the order of σ and E := C K (σ ). Clearly V is a vector space over E and T s GL(V ) GL(V E ). Let µ(x) E[x] be the minimal polynomial of T s on V E. We claim that µ(x) is irreducible. For, if g(x) E[x] is a proper nontrivial
3 [3] Semifields arising from irreducible semilinear transformations 335 divisor of µ(x), then Ker g(t s ) is a proper nontrivial subspace of V E. Since both K and T commute with g(t s ), they leave invariant the K -space Ker g(t s ), contrary to the irreducibility of T on V. STEP 2. Let µ (x) K [x] be an irreducible factor of µ(x). Then for some t s, µ(x) = t i= µ i (x) where the polynomials µ i (x) := µ σ i (x) K [x] are distinct and irreducible. For i t, let V i := Ker(µ i (T s )). Then (3.2) holds with T (V i ) = V i+ (the subscripts are mod t), and T s has minimal polynomial µ i (x) on (V i ) K. Moreover, m := T s V is K -linear with irreducible minimal polynomial µ (x) K [x], so that L := K [m ] is a subfield of End(V ) and V is a vector space over L. We always let v denote an arbitrary nonzero vector of V. We have T t (km v ) = k σ t T t (T s (v )) = k σ t T s (T t (v )) = k σ t m T t (v ) for k K. It follows that T t V is ρ-semilinear on V for an automorphism ρ of L = K [m ] that coincides with σ t on K, fixes m = T s and hence has the same order s/t as σ t. STEP 3. Most of the proof now focuses on the semilinear transformation T := T t V of V, rather than on T and V. The map T acts irreducibly on the K -space V. For, let W be a nonzero T -invariant subspace of V. Then W i := T i (W ) is a subspace of V i for i t, and T (W t ) = T t (W ) = T (W ) = W. By (3.2), W W t is a nonzero T -invariant subspace of V, and hence W = V, as required. STEP 4. By Step 2, T is semilinear on (V ) L with associated field automorphism ρ of order n := s/t. The polynomial algebra L[T ] (see [4]) is not commutative if ρ. This leads us to consider the set R of polynomials f (x) = d x j f j with f j L, using the twisted product x j a = a ρ j x j for a L. Then R is a (noncommutative) L-algebra having L[T ] as a homomorphic image under the substitution x T. Jacobson [4] viewed V as an R-module, but we will not need this point of view. We only need to know that each f R has a degree in the usual manner, and that f (T )(v ) = d T j f j(v ), where T j f j is a composition of additive maps on V. Then f (T ) is an additive map on V, but it need not be K -semilinear. STEP 5. Let f (x) = d x j f j R, f j L, f d =, have minimal degree d such that f (T )(V ) =. Then d n since (T n m I )(V ) = (T s m I )(V ) = (by the definition of m in Step 2). We claim that d = n; in fact we will show that f (x) = x n m. Take a L lying in no proper subfield, so that a a ρ j for < j < n. Consider g(x) := a ρd f (x) f (x)a R. On the one hand, g(t )(v ) = a ρd f (T )(v ) f (T )(av ) = a ρd =. On the other hand, calculating in R we find that g(x) = d (a ρd x j ) f j d x j ( f j a) = d x j (a ρd j a) f j
4 336 W. M. Kantor and R. A. Liebler [4] has degree < d since a ρd d a =. Now g(t )(V ) = and our choice of f (x) imply that (a ρd j a) f j = for j < d. Then f j = for < j < d (since a ρd j a), so that f (x) = x d + f. If d < n then a ρd a, so that f =, whereas T d(v ). Thus, d = n and f (x) = x n + f. Finally, since ( f + m )(V ) = we have f = m, as claimed. STEP 6. We next claim that V has dimension one as a vector space over L. Since T acts irreducibly on V by Step 3, it suffices to exhibit a -dimensional subspace of (V ) L fixed by T. By Step 2, m F := C L (ρ), where [L : F] = ρ = s/t = n. Consequently, if N L/F : L F is the norm map, then there exists an element a L such that N L/F (a) := n a ρ j equals m. Since h(x) := n (ax) j R has degree less than n, by Step 5 we have h(t )(V ). Let v V with w := n (at ) j (v). Then and hence (at ) n (v) = N L/F (a)t n (v) = N L/F(a)m v = v, n (at )(w) = (at ) j (v) + (at ) n (v) = n (at ) j (v) + v = w. Thus, T (Lw) = Lw, so that Lw is the required T -invariant -space over L, and hence dim(v ) L =. STEP 7. Finally, we need to show that the action of T on V determines T up to GL(V )-conjugacy. For, if B := {v i i =,..., d} is a K -basis of V and v i j := T j (v i ), then {v i j i =,..., d} is a K -basis of V j for j t. If A is the matrix of T = T t V with respect to B then v i v 2i v ti Av i, i d, uniquely describes T up to GL(V )-conjugacy. Observe that, in the notation of Steps and 2, K = C K (σ ) s = E nt, so V = L t = ( K deg µ ) t = E nt2 deg µ. PROOF OF THEOREM.. We are given a vector space V of size n = p r over the prime field GF( p). We will imitate the preceding proposition in order to construct semilinear transformations over subfields of End(V ) that include all irreducible ones but also include many others. Thus, we will need a decomposition (3.2), a subfield L of End(V ) implicit in the statement of Proposition 3., a field K, automorphisms of K and L (see Step 2 of the proposition), and a semilinear transformation T = T t V on V.
5 [5] Semifields arising from irreducible semilinear transformations 337 Choose a factorization r = te with e > and t e; the number of these factorizations is the number τ(r) of positive divisors of r other than. Fix a decomposition (3.2) of V into subspaces V i of size p e. Fix a subfield L = GF(p e ) of End(V ), so that V is an L-vector space. Given e, all such decompositions and fields are GL(r, p)- conjugate. Choose a subfield K GF(p) of L. Choose σ Aut(L) such that σ K has order divisible by t. Let ρ := σ t. (Thus, σ := σ K Aut(K ) and ρ Aut(L) satisfy σ t = ρ K, as required in Step 2 of the proof of Proposition 3.. According to that step we should also require that σ t = ρ, but we will ignore this restriction in our estimates.) Extend the action of K from V in order to make V and all V i vector spaces over K. All such extensions are GL(r, p)-conjugate. Choose l L, and let T End(V ) be v lv ρ, v V (see Proposition 3.). We can restrict the choice of l as follows. If M a : v av, a L, then M a T M a : v la ρ v ρ. Since we require different conjugacy classes of transformations T, we can restrict l to a set (e, ρ) of L /(L ) ρ = C L (ρ) = p e/ ρ coset representatives of (L ) ρ in L. Up to conjugacy in GL(r, p), the choices made above uniquely determine T = T t V, and hence also T by the last part of Proposition 3.. (However, we emphasize that a σ -semilinear map obtained in this manner need not be irreducible on V K.) Thus, the number of GL(r, p)-conjugacy classes of pairs K, T, with T an irreducible K -semilinear transformation that is not linear is at most (e, σ t ) (#K L, σ K ). (3.3) e r,e σ There are τ(r) choices for e and L, and then at most e choices for σ, at most τ(e) subfields K, and p e/ ρ elements in (e, ρ), where again ρ = σ t. Clearly, p e/ ρ dominates the corresponding term in (3.3). This is at most p r/3 unless σ has order 2 and either (i) L = p r, t =, ρ = σ has order 2 and (e, ρ) = p r/2 ; or (ii) L = p r/2, t = 2, ρ =, σ = 2 and (e, ρ) = p r/2. Possibilities (i) and (ii) together contribute at most 2(p r/2 )(τ(r) τ(r/2)) to (3.3). Then (3.3) is easily bounded as required in the theorem if r is not too small, leaving a few cases to be handled by a slightly more detailed and tedious examination of (3.3). 4. Concluding remarks We conclude with some elementary observations concerning the semifields S T and our arguments.
6 338 W. M. Kantor and R. A. Liebler [6] REMARK 4.. Note that π kt = πt, π T +k I = πt and π T = π T for all k K, since S kt = S T, S T +k I = S T and S T T d = S T. Thus, as in the desarguesian case, there are isomorphisms among the planes π T that do not arise from conjugate semilinear transformations. REMARK 4.2. As in Section 2, if we fix e V then we obtain a presemifield operation on V from S T via a b = g(a)(b), a, b V, using the additive isomorphism g : V S T defined by g(a(e)) = A for A S T. Then A(e) v = A(v) for all A S T, v V, gives a simple description of our operation. In fact, this turns V into a semifield with identity element e, since e v = I (e) v = I (v) and A(e) e = A(e) for all v and A. REMARK 4.3. It is straightforward to extend the action of L in Proposition 3. from V to all of V so as to make all V i into -dimensional L-spaces. However, as has been pointed out to us by Dempwolff via an example [3], there can be irreducible semilinear transformations over K that are not semilinear over any such extension field L. Nevertheless, a simple way to obtain a candidate for an irreducible σ -semilinear map on a vector space V over a field K is to use σ -semilinearity together with the requirement T : v v 2 v t mv (4.) for some basis {v,..., v t } of V and some m K. If t > in (4.), it is easy to check that the corresponding semilinear map has no invariant -space if and only if m / K +σ + +σ t. In this case, if t = 2 then the corresponding semifield was discovered by Knuth [7]. REMARK 4.4. Similarly, we can obtain many irreducible semilinear transformations by assuming σ -semilinearity in (4.). PROPOSITION 4.2. Let V be a vector space over K with basis v,..., v t, and let σ Aut(K ) and ρ = σ t. If m K with m σ j / K ρ for j < t, then (4.) defines an irreducible σ -semilinear transformation on V with associated field automorphism σ. PROOF. Suppose that W is a nonzero T -invariant subspace of V. Let t k i v i W, k i K, with the minimum number of k i. Using T we may assume that k =. By (4.) and the fact that W is T t -invariant, ( t ) t t T t k i v i m k i v i = (k ρ i mσ i v i k i mv i ) 2 lies in W and has smaller support, and hence is zero. Then k ρ i mσ i = k i m for 2 i t. If some such k i then m σ i = /(k ρ i ), contradicting our condition on m.
7 [7] Semifields arising from irreducible semilinear transformations 339 Thus, v W. Applying T shows that all v i W, so that W = V. REMARK 4.5. We conclude with a very elementary but weaker version of Theorem. (compare to [6, Theorem 6.2]) having a less informative proof. PROPOSITION 4.3. Given a vector space V of size n over a prime field GF(p), there are fewer than n log 2 p n conjugacy classes of pairs (K, T ) consisting of a field K End(V ) over which V is a vector space and an irreducible semilinear transformation T on V K. PROOF. Let d = dim K V. Let T be an irreducible σ -semilinear transformation of V K. Fix a nonzero vector v. Then {T i (v) i < d} is a basis of V (in Section 2 we saw that d k i T i (v) =, k i K, implies that all k i = ). Write T d (v) = d k i T i (v) with k i K. Since T i (kv) = k σ i T i (v) for each i and each k K, the k i completely determine T. Thus, T is determined by the following choices: a field K = GF(p e ) over which V is a vector space, an automorphism σ of K, and a choice of d = r/e elements k i K, where V = p r. There are at most r divisors e of r, at most e choices for σ, and then V choices for the k i. Choosing a K -basis of V amounts to conjugating in GL(V K ) and hence in GL(r, p). Consequently, the number of GF( p)-conjugacy classes of pairs (K, T ) is less than rr V = V log 2 p V, as required. Unlike in the proof of Proposition 3., this argument used all K d = p r possible d-tuples (k,..., k d ), which is independent of the choice of K and σ. Acknowledgement We are grateful to U. Dempwolff for very helpful comments. References [] P. Dembowski, Finite Geometries (Springer, Berlin, 968). [2] U. Dempwolff, Normalformen semilinearer operatoren, Math. Semesterber. 46 (999), [3], Private communication, 28. [4] N. Jacobson, Pseudo-linear transformations, Ann. Math. 38 (943), [5] V. Jha and N. L. Johnson, An analog of the Albert Knuth theorem on the orders of finite semifields, and a complete solution to Cofman s subplane problem, Algebras Groups Geom. 6 (989), 35. [6] W. M. Kantor, Finite semifields, in: Finite Geometries, Groups, and Computation, Proc. Conf., Pingree Park, CO, September 25 (eds. A. Hulpke et al.) (de Gruyter, Berlin, 26), pp [7] D. E. Knuth, Finite semifields and projective planes, J. Algebra 2 (965), WILLIAM M. KANTOR, Department of Mathematics, University of Oregon, Eugene, OR 9743, USA kantor@math.uoregon.edu ROBERT A. LIEBLER, Department of Mathematics, Colorado State University, Fort Collins, CO 8523, USA liebler@math.colostate.edu
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