Inner Automorphisms of Finite Semifields

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1 Note di Matematica Note Mat. 29 (2009), supp. n. 1, ISSN , e-issn Note DOI /i v29n1suppp231 di Matematica 29, n. 1, 2009, DOI /i v29n1suppp Università de Saento Inner Automorphisms of Finite Semifieds G. P. Wene Department of Mathematics, University of Texas at San Antonio, One UTSA Circe, San Antonio, Texas Abstract. Unike finite fieds, finite semifieds possess inner automorphisms. A further surprise is that even noncommutative semifieds possess inner automorphisms. We compute inner automorphisms and automorphism groups for semifieds quadratic over the nuceus, the Hughes-Keinfed semifieds and the Dickson commutative semifieds. Keywords: semifieds, inner automorphisms. MSC 2000 cassification: primary 17A35, secondary 17A36 1 Introduction Finite nonassociative division rings were introduced in 1905 by L. E. Dickson [7]. Current interest is driven by the fact that the finite panes of Lenz-Barotti type V.1 (transation panes) are precisey the panes coordinatizabe by division rings which are not fieds (see Biiotti, Jha and Johnson [2], Hughes and Piper [14]). Readers interested in the history of semifieds are refered to the artices Abert [1], Knuth [18], Keinfed [17], Cordero and Wene [6] and Kantor [15]. We wi use the term semified to refer to a not necessariy associative division ring. A finite semified [18] is a finite agebraic system containing at east two distinguished eements 0 and 1. A finite semified possesses two binary operations, addition and mutipication, designated in the usua notation and satisfying the foowing axioms: (i) (,+) is a group with identity 0. (ii) If a,b and ab = 0 then a = 0 or b = 0. (iii) If a,b,c then a(b+c) = ab+ac and (a+b)c = ac+bc. (iv) The eement 1 satisfies the reationship 1 a = a 1 = a for a a. It is easiy seen that there are unique soutions to the equations ax = b and xa = b for every nonzero a and every b in. It aso foows easiy that addition is commutative. In fact it can be shown that is a vector space over some

2 232 G. P. Wene prime fied GF(p) and that has p n eements where n is the dimension of over F, see [18]. Litte is known of the automorphism groups of finite semifieds. Both Dickson [8] and Menichetti [19, 20] made partia determinations of the automorphisms group of semified three dimensiona over a finite fied not of characteristic two. Keinfed [16] and Knuth [18] computed the automorphism group of each of the 23 isomorphism casses of 16-eement semifieds; Burmester[3] showed that there are n isomorphism casses of Dickson commutative semifieds of order p 2n, p 2, each of these semifieds has 2n automorphisms and determines the structure of these automorphisms; Zemmer [21] used automorphisms to determine the existance of subsemifieds. We begin with an examination of Knuth s System W [18]. 1 Exampe (Knuth s System W.). This semified is isomorphic to Keinfed s System T-35 [16]. Let F 4 be the four-eement fied with eements 0,1,ω and ω 2 (= ω + 1). The eements of System W are of the form a + λb where a,b F 4. Addition and mutipication are defined in terms of the addition and mutipication of F 4. (x+λy)+(u+λv) = (x+u)+λ(y +v) and (x+λy)(u+λv) = (xu+ωy 2 v)+λ(yu+x 2 v). This system has three automorphisms. These automorphisms are a inner and are given by Φ i (x+λy) = x+λω i y, i = 1,2,3. There is a unique subring of order 4 that is generated by ω and is the nuceus. Each of these automorphisms is inner. Φ 1 (x+λy) = [ω(x+λy)]ω 2 = x+λωy Φ 2 (x+λy) = [ ω 2 (x+λy) ] ω = x+λω 2 y Φ 3 (x+λy) = [1(x+λy)]1 = x+λy Knuth s System W is quadratic over its nuceus in the sense of Hughes and Keinfed [12]. We wi show that a semifieds quadratic over the nuceus possess inner automorphisms and wi compute the automorphism groups. The arguments appy to a arger cass of semifieds that incude Hughes-Keinfed semifieds and the Dickson commutative semifieds.

3 Inner Automorphisms of Finite Semifieds 233 We begin with some preiminaries. This is foowed by a cose ook at the automorphisms groups of the Hughes-Keinfed semifieds. If is a Hughes- Keinfed semified of order p 4, the automorphism group is competey determined. We construct a Dickson commutative semifieds that possess inner automophisms. The concusion gives severa directions for continued research. 2 Preiminaries A too used to study the the associativity of finite semifieds and nonassociative rings in genera is the associator of eements a, b and c: (x,y,z) = (xy)z x(yz). The three semi-nucei of a semified are defined in terms of associators and refect the rich structure of finite semifieds. The eft nuceus N is the set of a eements d in such that (d,x,y) = 0 for a x,y. The midde nuceus N m and the right nuceus N r are defined anaogousy. The intersection of the three semi-nucei of is caed the nuceus; the center, denoted by Z, refers to the set of a n in the nuceus N such that nx = xn for a x. A set W of eements of A is caed a weak nuceus if (a,b,c) = 0 whenever any two of a,b,c are in W. The nuceus wi aways be a subring of the weak nuceus. If is a finite semified, any one of the above nucei wi be a fied and may be considered as a eft vector space over N, N m, N and Z and a right vector space over N m, N r, N and Z. In a commutative semified, the eft nuceus is the right nuceus; this semi-nuceus is contained in the midde nuceus. The midde nuceus of a commutative semified is aways a weak nuceus. If the dimension of A over it s weak nuceus is two, we say that is quadratic over a weak nuceus. These semifieds have been investigated by Hughes and Keinfed [12], Knuth [18], Cohen and Ganey [4] and Ganey [11]. An automorphism of a semified is a bijection Θ : such that Θ(x+y) = Θ(x) +Θ(y) and Θ(xy) = Θ(x)Θ(y) for a a,b in. We wi denote by Aut( ) the group of automorphisms of. Automorphisms of semifieds wi be denoted by capita Greek etters and automorphisms of fieds wi be denoted by ower case Greek etters. The set S of a eements s of such that Θ(s) = s wi form a subring of ; if is a finite semified the set S of eements fixed by Θ wi be a semified. If Θ 2 = id, the identity automorphism, then the set S wi be caed the symmetric eements; if the characteristic is not two, the set K = {s : Θ(x) = x} wi be caed the set of skew eements. An automophism Θ of is caed an inner automorphism if there is an eement m with eft inverse m 1 (m 1 m = 1) such that Θ(x) = (m 1 x)m for a x in. We wi denote the inner automorphism x (m 1 x)m by Θ m.

4 234 G. P. Wene Ceary if m is a nonzero eement of a weak nuceus then m 1 r = m 1 = m 1 and Θ m (x) = (m 1 xm) = (m 1 x)m. The eements fixed by an inner automorphism wi generate a subsemified of the semified. If Θ m is an inner automorphism of and Φ an arbitrary automorphism of, then Φ 1 Θ m Φ wi be an inner automorphism of. Since the mapping x (m 1 x)m wi aways be an isomorphism of the additive group of, we need ony to determine if this mapping is an isomophism of the mutipicative structure of. 2 Lemma. Let denote a finite semified with nuceus N. If m N then the mapping Θ defined by Θ m (x) = m(xm 1 ) for a x in is an inner automorphism of. Proof. Ceary Θ m is a bijection. Let x,y then [(mx)m 1 ][(my)m 1 ] = (mx)[m 1 [(my)m 1 ]] = (mx)[[m 1 (my)]m 1 ]] = (mx)[ym 1 ] = m(x[ym 1 ]) = m([xy]m 1 ) = [m(xy)]m 1. It foows immediatey that If the inner automorphism group of a finite semified is trivia then N Z; if the semified has no proper subsemifieds the ony inner automorphism is the trivia automorphism. 3 Theorem. Let denote a finite semified with nuceus N. If Θ m (x) = (m 1 x)m defines an inner automorphism for some m, then so does Θ nm (x) = [(m 1 n 1 )x](nm) for each nonzero n N. Proof. By the previous Lemma, Θ n defines an inner automorphism for a n N. If Θ m (x) = (m 1 x)m defines an automorphism then so does Θ m Θ n. Θ m Θ n (x) = Θ m ((n 1 x)n) = ( Θ m (n 1 )Θ m (x) ) Θ m (n) = [(n 1 m 1 )x](nm) = Θ nm 4 Theorem. Let Θ m define an automorphism of the semified and et a,b be nonzero eements of the nuceus. Then Θ am and Θ bm define the same automorphism if and ony if ab 1 Z.

5 Inner Automorphisms of Finite Semifieds 235 Proof. Suppose Θ am = Θ bm. Then, for a x, [(m 1 a 1 )(x)](am) = [(m 1 b 1 )(x)](bm) [(m 1 a 1 )(x)]a = [(m 1 b 1 )(x)]b [m 1 (a 1 x)]a = [m 1 (b 1 x)]b m 1 [(a 1 x)a] = m 1 [(b 1 x)b] (a 1 x)a = (b 1 x)b xab 1 = ab 1 x 5 Coroary. Let denote a finite semified with nuceus N. The eements m,n N define the same inner automorphism of if and ony if n 1 m Z. In this case the eements of the nuceus determine ( N 1)/( N Z 1) inner automorphisms. 3 The Hughes-Keinfed Semifieds We begin this section a theorem of Hughes and Keinfed [12]. 6 Theorem (Hughes and Keinfed [12]). Let R be a not associative division ring which is a quadratic extension of a Gaois fied F, and suppose F is contained in the right and midde nucei of R. Then R must be isomorphic to a ring S constructed as foows: Let S be a vector space of dimension 2 over F, having a basis 1,λ and mutipication defined by (x+λy)(u+λv) = (xu+δ 0 y σ v)+λ(yu+x σ v +δ 1 y σ v), where σ is an arbitrary non-identity automorphism of F and δ 0,δ 1 in F are subject ony to the condition that w 1+σ +δ 1 w δ 0 = 0 have no soution for w in F. Conversey, given F,σ,δ 0,δ 1, satisfying the above conditions, then S wi satisfy the conditions on R. We wi imit our discussion to those Hughes-Keinfed semifieds for which δ 1 = 0 and wi write the product as (x+λy)(u+λv) = (xu+δy σ v)+λ(yu+x σ v). Ceary (a+λb) (a λb) defines an automorphism of these semifieds whenever the characteristic is not two. Motivated by exampe 1, we ask when does the mapping x+λy [f 1 (x+ λy)]f, where f is a nonzero eement of F, define an automorphism of the semified?

6 236 G. P. Wene 7 Theorem. Let be a Hughes-Keinfed semified and θ : be defined by θ(x) = [f 1 x]f, x, where f F. Then θ is an automorphism if and ony if f σ2 = f. Proof. [ ( f 1) (x+λy)]f = x+λy ( f 1) σ f θ([x+λy)][u+λv]) = xu+δy σ v +λ(x σ v +yu) ( f 1) σ f and ( ( x+λy f 1 ) σ )( ( f u+λv f 1 ) σ ) f = xu+δy σ v [( f 1) σ ] σ ( f f 1 ) σ f + λ(x σ v +yu) ( f 1) σ f. δu σ v = δu σ v [( f 1) σ ] σ ( f f 1 ) σ f 1 = ( f 1) σ 2 f f σ2 = f 8 Coroary. If the Hughes-Keinfed semified has a subfied F 0 F fixed pointwise by σ 2 then x [f 1 x]f defines an automorphism of. Those semifieds with the argest possibe nucei are the semifieds quadratic over the nuceus; Hughes and Keinfed [12] computed these semifieds. 9 Theorem (Hughes and Keinfed [12]). Let R be a not associative division ring which is a quadratic extension of a Gaois fied F, and suppose F is contained in the nuceus of R. Then R must be isomorphic to one of the rings S of theorem 6 with the additiona stipuation that σ 2 = I and δ 1 = 0 conversey, a such S satisfy the conditions on R. 10 Theorem. Let be a semified quadratic over a nuceus isomorphic to the Gaois fied GF(q 2 ). Then the eements of the nuceus of determine q+1 inner automorphisms. Proof. There are q 2 1 nonzero eements in the nuceus and q 1 nonzero eements in N Z. These eements determine ( q 2 1 ) /(q 1) = q +1 inner automorphisms. The cassification of semifieds of order p 4 has yet to be competed; a nice beginning is Cordero [5]. If the order of a semified quadratic over its nuceus is p 4, for some prime p 2, the automorphism group of the semified is easiy computed. The characteristic case is exampe 1.

7 Inner Automorphisms of Finite Semifieds Theorem. Let be a semified quadratic over its nuceus, of order p 4. If Φ : is an automorphism then Φ(x + λy) = x k + λ(sy k ) where s is a nonzero eement of GF(p 2 ) and k is 1 or p. Furthermore, s is a soution of δ k = δx σ x. Proof. The condition impies that F is the fied GF(p 2 ). If Φ : is an automorphism then Φ : N N is a fied automorphism. Since N is isomorphic to the finite fied GF(p 2 ), Φ(n) = n k, where k is 1or p. We must know that what Φ does to λ. Suppose that Φ(λ) = r +λs where s is a nonzero eement of GF(p 2n ). Let α GF(p 2n ) then Φ(αλ) = Φ(α)Φ(λ) = α k (r +λs) = α k r +λ((α kσ )s) = Φ(λα σ ) = (r +λs)(α kσ ). Equating components, we find that α k r = α kσ r. If r 0, σ is the identity automorphism. Hence Φ(λ) = λs, Φ ( λ 2) = Φ(δ) = δ k = (λs)(λs) = [δs σ s]. We must have δ k = δs σ s. If k = 1, then δ = δs σ s. Now s σ s must be 1. There are exacty p n +1eements s such that s σ s = 1. There are p + 1 automorphisms a + λb a+λsb; these automorphisms form a subgroup isomorphic to the additive group Z p+1. If k = p we have δ p = δs σ s. Now s σ s is fixed by Φ and must be 1. Thus δ σ = δ and 1 is a square in GF(p). There are exacty p+1 eements s such that s σ s = 1. In the atter case the automorphism group is not commutative. Let Φ s and Ψ t be automorphisms of defined by Φ s (a+λb) = a+λsb and Ψ t (a+λb) = a p + λtb p where s p+1 = 1 and t p+1 = 1. Then Ψ t (Φ s (a + λb)) = a p + λts p b p and Φ s (Ψ t (a+λb)) = a p +λstb p. 12 Lemma. Let k be a nonsquare eement of GF(p n ) and m an eement of the extension fied GF(p 2n ) such that m 2 = k. Then m is a nonsquare in GF(p 2n ) if and ony if 1 is a square in GF(p n ). Furthermore, m p = m. Proof. Let k be a nonsquare eement of GF(p n ) and m an eement of the extension fied GF(p 2n ) such that m 2 = k. Suppose m is a square in GF(p 2n ) and m = (α+βm) 2. Then m = α 2 +β 2 k+2αβm. We must have α 2 +β 2 k = 0

8 238 G. P. Wene and 2αβm = 1. Soving these equations for α, we find that 4α 4 = k. Hence if both k and k are nonsquares in GF(p), m wi be a nonsquare in GF(p 2 ). Since p is odd, m p = zm where z GF(p). Then m p2 = m = zm p = z 2 m and z = 1. The above emma tes us that we can aways find a nonsquare eement m in GF(p 2n ) such that m pn = m if 1 is a square in GF(p n ). 13 Exampe. Let F be the fied GF(25) isomorphic to GF(5)[m] where m 2 = 2. Then m is a nonsquare in F such that m 5 = m. We construct the semified as before using δ = 1+2m. Since δ 5 δ, the automorphism group consists of the six inner automorphisms generated by the automorphism Φ(a+λb) = a+λb(3+2m) where (3+2m) 3 = 1. If we use δ = m, we get a 12-eement automorphism group. The group is generated by the automophisms Φ as above and the automorphism Ψ(a+λb) = a 5 +λb 5 (1+m). The eements Φ and Ψ satisfy Ψ 2 = Φ 3 and ΨΦΨ 1 = Φ Remark. The congruence x (mod p) has a soution for the prime p ony if p is of the form 4n+1 (Dickson [10]). Some primes p for which 1 is a square in the finite fied GF(p) are p = 5,13,17,29,41,53,61,73,89 and 97. Let be a Hughes Keinfed semified that is not necessariy quadratic over its nuceus. The eft inverse λ 1 of the eement λ is and ( ) 1 σ 1 λ 1 = λ δ [ ( λ 1 ) (a+λb)]λ = a σ +λb σ. 15 Theorem. Let be a Hughes-Keinfed semified and θ λ : be defined by θ λ (x) = [λ 1 x]λ for x. Then θ λ is an automorphism if and ony if δ σ = δ. In particuar, is not quadratic over its nuceus. and Proof. θ λ ([a+λb)][c+λd]) = a σ c σ +δ σ b σ2 d σ +λ(a σ2 d σ +b σ c σ ) (a σ +λb σ )(c σ +λd σ ) = a σ c σ +δb σ2 d σ +λ(a σ2 d σ +b σ c σ ). Equatingcomponents yeidsδ σ = δ. Were tobe quadratic over its nuceus, this woud force δ to be a square in the fied F.

9 Inner Automorphisms of Finite Semifieds Exampe. Let F be the fied GF(5 3 ). Since 2 is a nonsquare in GF(5), it remains a nonsquare in GF(5 3 ). Construct the Hughes-Keinfed semified with product (x+λy)(u+λv) = (xu+2y 5 v)+λ(yu+x 5 v). The automorphism θ λ (x + λy) x 5 + λy 5 generates a cycic subgroup of three automophisms. If Φ : is the automorphism (x + λy) x λy then Φ θ λ generates a cycic subgroup of order six. 4 The Dickson Commutative Semifieds The Dickson commutative [9] semifieds are the ony commutative semifieds quadratic over a weak nuceus. The definitive study of the Dickson commutative semifieds is the paper by Burmester [3]. Surprisingy, the Dickson commutative semifieds possess inner automorphisms. The ony inner automorphims of a finite fied is the trivia automorphism. The rea quaternions are an infinite, associative, noncommunitive division ring that permits inner automorphisms. We wi construct a subcass of the Dickson commutative semifieds that have nontrivia inner automorphisms. 17 Exampe. Let F be the fied GF(p n ), p 2 and n 2. The eements of are of the form a + λb where a,b F. Addition and mutipication are defined in terms of the addition and mutipication of F, an automorphism σ of F and an eement δ F that is a nonsquare in F. The addition is given by and the mutipication by (a+λb)+(c+λd) = (a+c)+λ(b+d) (a+λb)(c+λd) = ac+δ(bd) σ +λ(ad+bc). Burmester [3] showed that the automorphisms of are given by Φ ij (a+λb) = a pi +λ(s ij b pi ),i = 0,1,...,n 1 and j = 1,2 where s ij is one soution of δ pi = δ(x 2 ) σ. He shows that there are n isomorphism casses of Dickson commutative semifieds of order p 2n, p 2; each of these semifieds has 2n automorphisms. We now derive an aternative description, in terms of inner automorphisms, for some of these automorphism groups. An obvious automorphism is the mapping (a+λb) (a λb).

10 240 G. P. Wene 18 Theorem. Let be a Dickson commutative semified, F the fied GF(p n ), p 2 and n 2 and δ a nonsquare eement of F. Then Φ(a+λb) = [ λ 1 (a+λb ] λ = a σ +λb σ defines an automorphism of if and ony if δ σ = δ. Proof. λ 1 = λ 1 δ then [ λ 1 (a+λb) ] λ = a σ +λb σ The mutipication property of the automorphism: Φ[(a+λb)(c+λd)] = Φ(ac+δ(bd) σ +λ(ad+bc)) = (ac) σ +δ σ (bd) σ2 +λ(ad+bc) σ (a σ +λb σ )(c σ +λd σ ) = (ac) σ +δ(bd) σ2 +λ(ad+bc) σ Equating components gives δ σ = δ. 19 Exampe. Let be a Dickson commutative semified, F the fied GF(5 5 ), and δ = 2. Since 2 is a nonsquare in GF(5) it remains a nonsquare in GF(5 5 ). The cycic automorphism group is generated Φ Ψ where Φ(a+λb) = a 5 + λb 5 = [ λ 1 (a+λb ] λ and Ψ(a + λb) = a λb. The eements of fixed by the automorphism Φ is the 25 eement fied GF(5)[λ]. There wi be ten automorphisms. 5 Concusion And Further Directions We have seen that a semifieds quadratic over the nuceus have ( nontrivia) inner automorphisms: if = p 4 for some prime p, the automorphism group can be competey determined. Any Hughes-Keinfed semified in which there is a subfied fixed pointwise by the automorphism σ has an inner automorphism as does any Hughes-Keinfed semified in which δ δ = δ. Our resuts can be used to produce many exampes. We determined a sufficient condition that a Dickson commutative semified have a (nontrivia) inner automophism. Much work remains to be done. The automorphism groups of the Hughes- Keinfed semifieds need to be computed. We need to find additiona exampes of semifieds with inner automorphisms. Dickson [8] discovered a certain famiy of three-dimensiona commutative nonassociative division agebras. Let F be any fied of characteristic 2. Let

11 Inner Automorphisms of Finite Semifieds 241 B,β,b be eements of F such that x 3 Bx 2 βx b is irreducibe over F. Define an agebra with basis 1,i,j by i 2 = j ij = ji = b+βi+bj j 2 = 4bB β 2 8bi 2βj. There has yet to emerge a comprehensive description of the automorphism of either the commutative or noncommutative semifieds 3-dimensiona over a finite fied. Compete the work started by Dickson and Menichetti. We wi use F K to denote the agebra that resuts from extending the base fied F to the fied K and are interested in the case where F K is a semified. What are the automorphism groups of the resuting semifieds F K? Does the obvious extension of the automorphism group aways work? The poynomia x 3 Bx 2 βx b wi continue to be irreducibe in a fied extensions K of degree prime to 3 and can be used to construct a commutative semified. If θ : is an automorphism and σ : K K an automorphism of the fied K then σ θ : F K F K wi be an automorphism. Ceary thepoynomiaw 1+pk +δ 1 w δ 0 usedtoconstructthehughes-keinfedsemified wi remain irreducibe in a fied extensions K of degree prime to p k +1; does the obvious extension of the automorphism group give the automorphisms group of F K? Knowing that the automorphism group has an eement of order 2 immediatey provides some knowedge of the mutipication of the agebra. What other interesting conditions can we impose on the automorphism group? There is much to do. References [1] A. A. Abert: Nonassociative agebras I. Fundamenta concepts and isotop y, Ann. Math., (2) 43, [2] M. Biiotti, V. Jha, N. L. Johnson: Foundations of Transation Panes, Pure and Appied Mathematics, Marce Dekker, New York, Base, 243 (2001), [3] M. V. D. Burmester: On the commutative non-associative division agebras of even order of L. E. Dickson, Rend. Mat. e App., V. Ser. 21 (1962), [4] S. D. Cohen, M. J. Ganey: Commutative semifieds, two dimensiona over their midde nuceui, J. Agebra 75 (1982), [5] M. Cordero: Semified panes of order p 4 that admit a p-primitive Baer coineation, Osaka J. Math. 28 (1991), [6] M. Cordero, G. P. Wene: A survey of finite semifieds, Discrete Mathematics 208/209 (1999),

12 242 G. P. Wene [7] L. E. Dickson: On finite agebras, Nachr. ges. Wiss. Göttingen, (1905), [8] L. E. Dickson: Linear agebras in which division is aways uniquey possibe, Trans. Amer. Math. Soc., 7 (1906), [9] L.E. Dickson: On commutative inear agebras in which division is aways possibe, trans. Amer. Math. Soc., 7 (1906), [10] L. E. Dickson: Eementary Number Theory, McGraw-Hi Book Company, Inc., New York, 1939, 484. [11] M. J. Ganey: Centra weak nuceus semifieds, Europ. J. Combinatorics, 2 (1981), [12] D. R. Hughes, E. Keinfed: Seminucear extensions of Gaois fieds, Amer. J. Math. 82 (1960), [13] D. R. Hughes: Coineation groups on non-desarguesian PanesII. Some seminucear division agebras., American J. Math. 82 (1960), [14] D. R. Hughes, F. C. Piper: Projective Panes, Springer Verag, New York, 1982, 291. [15] Kantor, W. M. (2006): In Finite semifieds, Groups and Computations,Proc. Of Conf. at Pingree Park, Co, Sept. 2005, Berin-New York: de Gruyter. [16] E. Keinfed: Techniques for enumerating Veben -Wedderburn systems, J. Assoc. Comp. Mach. 7 (1960), [17] Keinfed, E. (1983): A history of finite semifieds, in Finite Geometries, Puman, Washington, 1981, New York: Dekker, New York. [18] D. E. Knuth: Finite semifieds and projective panes, J. Agebra 2 (1965), [19] G. Menichetti: On a Kapansky conjecture concerning three-dimensiona division agebras over a finite fied, J. Agebra 47 (1977), [20] G. Menichetti: Agebre tridimensionai su un campo di Gaois, Ann. Mat. Pura App., 97 (1973), [21] J. L. Zemmer Jr.: On the subagebras of finite division agebras, Canadian J. Math. 4 (1952), Chicago, 1943.