Problems on structure for quasifields of orders 16 and 32 Текст научной статьи по специальности «Математика»

УДК 512.554

Problems on Structure for Quasifields of Orders 16 and 32

Vladimir M. Levchuk* Polina K. Shtukkertt

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 10.05.2014, received in revised form 10.06.2014, accepted 25.06.2014 Well-known method of the construction of finite projective translation planes (analogously, semifield planes) uses their correspondence with quasifields (resp., semifields). We distinguish certain questions on the structure of any finite quasifield (possible maximal subfields, the property of cyclicity of multiplicative loop of non-zero elements and possible orders of elements). In the present paper we discover some anomalous properties of finite quasifields of small even orders.

Keywords: projective translation plane, quasifield, semifield, multiplicative loop, orders of elements.

A ring S = (S, +, o) with the identity e = 0 is said to be a semifield (or "quasitelo according to [1, II. 6.1]), if S* = (S\ {0}, o) is a loop, i.e., for any a e S* and b e S each equation aox = b and y o a = b is uniquely solvable in S. For finite S the weakening of two-sided distributivity to one-sided one gives a quasifield [2,3]. It is well-known that there exists its unique minimal subfield of a prime order p and hence the order |S| is p-primary.

The construction of proper (or not being a field) quasifields is closely related to construction of non-Desargues projective translation planes and from the middle of last century it is based on computer calculations. Unlike finite fields, finite quasifields and semifields are poorly studied [4].

By the order |v| of element v of the loop we shall call (generalizing the notion of the order of a group element) the smallest integer m > 1 such that at least one m-th degree of element v at all possible positioning of brackets shall be equal e; the order is infinite when such m does not exist. The set of orders of all elements of a loop is called a spectrum.

For a finite proper quasifield the first author wrote down the following questions.

(A) Enumerate maximal subfields and their possible orders.

(B) What loop spectrums S* of finite semifields and quasifields are possible ?

(C) Enumerate finite quasifields, in particular, semifields where the loop S* is not singly-generated.

It remains open still the hypothesis of Wene on right primitive of semifield of order > 32 ( [5,6]): Is it true that the right-ordered degrees of a fixed element give all elements in S*? See also N.D. Podufalov [7] and his questions 9.43, 10.48, 11.76, 11.77 and 12.66 in [8].

Smallest even orders of non-Desargues projective semifield planes and translation planes are the same (unlike odd orders) and they are equal to 16, by virtue of [9] and [10]. Such planes are enumerated in [11-13] (see also [14,15]), and of order 32 — in [12,16]. Up to isotopisms, corresponding semifields and quasifields exhaust all ones of the same orders. For their P.K.Shtukkert have studied the issues (A)-(C) in the case of semifields, see [17] and Section 2 below. In the present paper we use the Kleinfeld's classification of semifields of order 16, up to isomorphisms.

* Vladimir M. Levchuk tPoli422@yandex .ru © Siberian Federal University. All rights reserved

According to [11], the number of isotopic classes and the number of isomorphic classes of proper semifields of order 16 is equal to 2 and 23, respectively. Theorems 2.1, 2.2 and Tab. 4 in Section 2 solve questions (A)-(C) for all semifields of order 16. Note that up to isomorphisms and anti-isomorphisms the number of such semifields is equal to 16. Using [16] we show that there exists a quasifield Q of order 16, which are the set-theoretic union of its 7 subfields of order 4; in particular, the loop Q* is not singly-generated and its spectrum is {1, 3} (Theorem 3.3 in Section 3).

Anomalous properties of proper semifields of order 32 are considered in Section 4.

1. The tie of quasifields and translation planes

The projective plane is defined as a set of points with certain subsets, which are called lines [18, Section 20.1]; ibid see definition of the plane order.

For the construction of a translation plane n of rank n it must be chosen a n-dimensional linear space W over a field F (coordinatized set), the outer direct sum of two copies of W,

V = W © W = {(x, y) | x, y e W},

and the spread p of additive group (V, +) such that V = M © N for any M = N from p. By definition, 1-dimensional subspaces in V are points of projective translation plane n = n(V, p), subgroups from p and their cosets are lines of n and, also, different cosets on the same subgroup have a unique general point (to) and all such singular points give a singular line [to] of n, [2].

Recall that the spread of an additive group is a set of its subgroups (components of spread), which have trivial pairwise intersections and their set-theoretic union gives whole group. The components of our spread p are n-dimensional subspace in V [3]. It is well-known the following lemma [19], where

V(a) = {(v, va) | v e W} (a e GL(W)), V(0) = (W, 0), V(to) = (0, W).

Lemma 1.1. Let us assume that V(0), V(to) e p. Then:

a) if M e p and M = V(0), V(to), then M = V(a) at unique a e GL(W) and, in particular, p = {V(a) | a e R* U {0}} U {V(to)} at R* = {a e GL(W) | V(a) e m};

b) if u, v e W \ {0}, then U = v at unique a e R*;

c) if T,p e R* and T = p, then t - p e GL(W).

Conversely if a subset R* in GL(W) satisfies b) and c), then p = {V(0), V(to)}U{V(a) | a e R*} is a spread of the group (V, +) such that V = M © N for any M = N from p.

Taking into account b) there exists a bijective mapping d : W ^ R* U {0} such that

0(v) = a (v e W \{0}, uff = v), 0(0) =0.

The totality R of a subset R* in GL(W) with the identity satisfying b), c) and the null map is said to be a regular set of plane n. Writing the vectors from W as coordinate rows and setting

x o y := x • 0(y) (x,y e W) (1)

we obtain a quasifield W = (W, +, o). If W is a semifield then n is called a semifield plane.

Definition 1.1. Quasifields (Si, +, o) and (S2, +, •} are called isotopic, if there exist isomorphisms F, G, H of additive groups Si ^ S2 such that

xF • yG = (x o y)H (x, y e Si).

It is well-known the following lemmas.

Lemma 1.2. (Albert, 1960) Two semifield planes are isomorphic if and only if the corresponding semifields are isotopic.

Lemma 1.3. The projective translation plane is Desargues if and only if the corresponding quasifield is a field.

Lemma 1.4. Coordinatized set is a field if and only if the regular set is subfield of a ring M(n, F) of all n x n-dimentions matrixes over F.

We now consider the structure of semifield planes of order 16. Any such plane n can be coordinatizate by 4-dimension space W over Z2. Choosing a regular set R = e(W) with the Z2-linear map в : W ^ M(4, Z2), which is identical onto (1,0,0,0). When (W, +, o) is a finite field with multiplication (1), by lemma 1.4, R is a subfield of order 16 in the ring M(4, Z2). Then R* is a cyclic group of order 15 which is generated by a matrix A e GL(4,2) with irreducible characteristic polynomial over Z2; for construction of such matrices we may use the natural normal form of matrices [20, Section 15.5]. In general we have

e(xb x2,x3,x4) = x1 • E + x2 • B + x3 • C + x4 • D (B, C, D e GL(4, Z2)).

By computer calculations we obtain exactly 19936 different sets {B, C, D} in which 336 cases give a field R = e(W). In other cases any set {B, C, D} uniquely defines a non-Desargues semifield plane n (together with R) and a semifield W with multiplication (1). All non-Desargues semifield planes of rang 2 over GF(4) are pairwise isomorphic [21]. In fact by methods [21] we may show that the enumeration of nonisomorphic non-Desargues semifield planes gives two cases:

/ x y z w \

w x + w z + w y + z + w

z z + w x + y + w y + w

\ y z y + w x J

x y z w

w x + z z + w y + w

z w x + y + z + w y + z + w

^ y + z + w z y + z x + w J

Taking into account (1) and lemma 1.2, we find, up to isotopism, exactly 2 proper semifields of order 16 with the multiplication, respectively,

(a, b, c, d) o (u, v, z, w) = (au + bv + cz + dw, av + bu + 6w + cz + cw + dz, az + bz + bw + cu + cv + cw + dv + dw, aw + bv + bz + bw + cv + cw + du); (2)

(a, b, c, d) o (u, v, z, w) = (au + bv + cz + dv + dz + dw, av + bu + bz + cw + dz, az + bz + bw + cu + cv + cz + cw + dv + dz, aw + bv + bw + cv + cz + cw + du + dw). (3)

We now obtain the following theorem which was earlier proved also with computer calculations, see E.Kleinfeld [11] and D.Knuth [12].

Theorem 1.1. There exist only 3 nonisomorphic semifield planes of order 16.

In the investigation of any quasifield the impotent role play the left, middle and right kernels. For any semifield S it is, respectively,

Nl(S) = {x e S | x o (y o z) = (x o y) o z, Vy, z e S}, Nm(S) = {y e S | x o (y o z) = (x o y) o z, Vx, z e S}, Nr (S) = {z e S | x o (y o z) = (x o y) o z, Vx, y e S}.

(x, y, z, w) =

(x, y, z, w) =

(x, y, z, w G Z2).

2. The construction of semifields of order 16

In 1960 E.Kleinfeld [11] obtained the classification of proper semifields of order 16, up to isomorphisms. It had been shown that one isotopic class has 18 pairwise nonisomorphic semifields Vi, V2, • • • , V18, and second has 5 pairwise nonisomorphic semifields T24, T25, T35, T45, T50.

Note that the constructed semifields with the multiplication (2) and (3) in Section 2 are isomorphic to semifields V7 and T25, respectively. We now investigate their structure.

Theorem 2.1. Let S be the semifield with multiplication (2), i.e., S ~ V7. Then:

(i) the minimal subfield Z2e of S is maximal;

(ii) the loop S* is generated by each nonidentity element;

(iii) the spectrum of loop S* is {1, 4, 5, 6}.

Proof. Firstly we find Cayley's table of loop multiplication S*, see Tab. 1. (In the table multiplications on identity element (1,0,0,0) is omitted.)

Further we denote by gk, the k—th degree (k ^ 1) of element g of semifield with the right (or the right-normalized) positioning of brackets. The Tab. 1 shows that the right k-th degrees (1 ^ k ^ 15) of each element

m1 = (0,1,1, 1), m2 = (1, 1, 0, 0), m3 = (1,1, 0,1), m4 = (1,1,1, 0)

give all elements of the loop S*. In particular, each element m; generates loop S* and (m,)15 = e.

Since for the element h = (0, 0,0,1) all products of length < 5 differ from e and h2 • h3 = e, so |h| = 5. On the other hand, for the element m1 all possible products of length < 5 also do not equal e and m2 • (m1 • (m1 )3) = e. Therefore |m1| = 6. Analogously we show that any element of the loop S * has the order ^ 6. The orders of all elements of the loop S * are given in Tab. 2.

In particular, the analog of group-theoretic Lagrange's theorem is not satisfied even for the orders of elements of the loop S*. Also we consider a table of left and right invertible elements.

The Tabs. 1 and 3 show that any nonidentity element of the loop S* is a suitable degree of some element of m; and, therefore, it generates the loop S*. In particular, S* is the right-cyclic or the right-primitive in terms of [6].

It is clear that for each element of any subfield left and right invertible elements are coincide. In according to Tabs. 2 and 3, such nonidentity elements are only elements (1,1,0,0) and (1,1,1,0) of order > 3. Consequently, the semifield S has no a subfield of order > 2. □

For the semifield with multiplication (3) it is true

Theorem 2.2. Let S be the semifield with multiplication (3),i.e., S ^ T25. Then:

(i) there exist exactly 2 maximal subfields Hi and H2, and |Hi| = |H2| = 4;

(ii) the loop S* is generated by each element from S \ {H1 U H2};

(iii) the spectrum of loop S* is {1, 3, 4, 5, 6}.

In fact, it was obtained the analogous structural description for all 23 proper Kleinfeld's semifields of order 16. It seems, up to isomorphisms and antiisomorphisms, there exist only 16 of proper semifields of order 16. More exactly it is proved

Theorem 2.3. Any proper semifield of order 16 up to isomorphisms is either one of 7 semi-fields Vi, V3, V4, V8, V11, V15, T25 or one of opposite semifields to them V6, V7, V5, V9, V14, V6, T50, respectively, or one of 9 semifields V2, V10, V12, V13, V17, V18, T21, T35, T45.

For any semifield W the opposite semifield is denoted by Wop. The following Tab. 4 gives the structure of proper semifields of order 16 up to isomorphisms and antiisomorphisms.

Table 1. Loop S * with multiplication

(0,0,0,1) (0,0,1,0) (0,0,1,1) (0,1,0,0) (0,1,0,1) (0,1,1,0) (0,1,1,1)

(0,0,0,1) (0,0,1,0) (0,1,0,0) (0,1,1,0) (1,0,1,0) (1,0,0,0) (1,1,1,0) (1,1,0,0)

(0,0,1,0) (0,1,1,1) (1,1,0,0) (1,0,1,1) (0,0,1,1) (0,1,0,0) (1,1,1,1) (1,0,0,0)

(0,0,1,1) (0,1,0,1) (1,0,0,0) (1,1,0,1) (1,0,0,1) (1,1,0,0) (0,0,0,1) (0,1,0,0)

(0,1,0,0) (1,1,1,1) (0,0,1,1) (1,1,0,0) (0,0,0,1) (1,1,1,0) (0,0,1,0) (1,1,0,1)

(0,1,0,1) (1,1,0,1) (0,1,1,1) (1,0,1,0) (1,0,1,1) (0,1,1,0) (1,1,0,0) (0,0,0,1)

(0,1,1,0) (1,0,0,0) (1,1,1,1) (0,1,1,1) (0,0,1,0) (1,0,1,0) (1,1,0,1) (0,1,0,1)

(0,1,1,1) (1,0,1,0) (1,0,1,1) (0,0,0,1) (1,0,0,0) (0,0,1,1) (0,0,1,1,) (1,0,0,1)

(1,0,0,1) (0,0,1,1) (0,1,1,0) (0,1,0,1) (1,1,1,0) (1,1,0,1) (1,0,0,0) (1,0,1,1)

(1,0,1,0) (0,1,1,0) (1,1,1,0) (1,0,0,0) (0,1,1,1) (0,0,0,1) (1,0,0,1) (1,1,1,1)

(1,0,1,1) (0,1,0,0) (1,0,1,0) (1,1,1,0) (1,1,0,1) (1,0,0,1) (0,1,1,1) (0,0,1,1)

(1,1,0,0) (1,1,1,0) (0,0,0,1) (1,1,1,1) (0,1,0,1) (1,0,1,1) (0,1,0,0) (1,0,1,0)

(1,1,0,1) (1,1,0,0) (0,1,0,1) (1,0,0,1) (1,1,1,1) (0,0,1,1) (1,0,1,0) (0,1,1,0)

(1,1,1,0) (1,0,0,1) (1,1,0,1) (0,1,0,0) (0,1,1,0) (1,1,1,1) (1,0,1,1) (0,0,1,0)

(1,1,1,1) (1,0,1,1) (1,0,0,1) (0,0,1,0) (1,1,0,0) (0,1,1,1) (0,1,0,1) (1,1,1,0)

(1,0,0,1) (1,0,1,0) (1,0,1,1) (1,1,0,0) (1,1,0,1) (1,1,1,0) (1,1,1,1)

(0,0,0,1) (0,0,1,1) (0,1,0,1) (0,1,1,1) (1,0,1,1) (1,0,0,1) (1,1,1,1) (1,1,0,1)

(0,0,1,0) (0,1,0,1) (1,1,1,0) (1,0,0,1) (0,0,0,1) (0,1,1,0) (1,1,0,1) (1,0,1,0)

(0,0,1,1) (0,1,1,0) (1,0,1,1) (1,1,1,0) (1,0,1,0) (1,1,1,1) (0,0,1,0) (0,1,1,1)

(0,1,0,0) (1,0,1,1) (0,1,1,1) (1,0,0,0) (0,1,0,1) (1,0,1,0) (0,1,1,0) (1,0,0,1)

(0,1,0,1) (1,0,0,0) (0,0,1,0) (1,1,1,1) (1,1,1,0) (0,0,1,1) (1,0,0,1) (0,1,0,0)

(0,1,1,0) (1,1,1,0) (1,0,0,1) (0,0,0,1) (0,1,0,0) (1,1,0,0) (1,0,1,1) (0,0,1,1)

(0,1,1,1) (1,1,0,1) (1,1,0,0) (0,1,1,0) (1,1,1,1) (0,1,0,1) (0,1,0,0) (1,1,1,0)

(1,0,0,1) (1,0,1,0) (1,1,1,1) (1,1,0,0) (0,1,1,1) (0,1,0,0) (0,0,0,1) (0,0,1,0)

(1,0,1,0) (1,1,0,0) (0,1,0,0) (0,0,1,0) (1,1,0,1) (1,0,1,1) (0,0,1,1) (0,1,0,1)

(1,0,1,1) (1,1,1,1) (0,0,0,1) (0,1,0,1) (0,1,1,0) (0,0,1,0) (1,1,0,0) (1,0,0,0)

(1,1,0,0) (0,0,1,0) (1,1,0,1) (0,0,1,1) (1,0,0,1) (0,1,1,1) (1,0,0,0) (0,1,1,0)

(1,1,0,1) (0,0,0,1) (1,0,0,0) (0,1,0,0) (0,0,1,0) (1,1,1,0) (0,1,1,1) (1,0,1,1)

(1,1,1,0) (0,1,1,1) (0,0,1,1) (1,0,1,0) (1,0,0,0) (0,0,0,1) (0,1,0,1) (1,1,0,0)

(1,1,1,1) (0,1,0,0) (0,1,1,0) (1,1,0,1) (0,0,1,1) (1,0,0,0) (1,0,1,0) (0,0,0,1)

Table 2. The orders of elements of the loop S *

y (1,0,0,0) (0,0,0,1) (0,0,1,0) (0,0,1,1) (0,1,0,0) (0,1,0,1) (0,1,1,0) (0,1,1,1)

|y| 1 5 5 5 5 5 5 6

y (1,0,0,1) (1,0,1,0) (1,0,1,1) (1,1,0,0) (1,1,0,1) (1,1,1,0) (1,1,1,1)

|y| 6 5 4 6 6 5 4

3. Quasifields of order 16

Kleinfeld [11] classified quasifields of order 16 with kernel of order 4 using computer calculations. Also he noted: "The problem of determining all such Veblen-Wedderburn systems turns out to be more difficult in the previous case and we abandon it in favor of determining all such division rings".

U.Dempwolff and A.Reifart (see works [13,16]) have completely classified translation planes of order 16. Up to isomorphisms, there are exactly 8 planes and the number of classes containing

Table 3. Right and left invertible elements in the loop S*

The element The left invertible The right invertible

(1,0,0,0) (1,0,0,0) (1,0,0,0)

(1,1,0,0) (1,1,1,0) (1,1,1,0)

(1,1,1,0) (1,1,0,0) (1,1,0,0)

(0,0,0,1) (0,1,1,0) (0,1,0,1)

(0,0,1,0) (0,0,1,1) (0,1,1,1)

(0,1,0,0) (0,1,1,1) (1,0,1,1)

(0,0,1,1) (1,0,1,0) (0,0,1,0)

(0,1,0,1) (0,0,0,1) (1,0,0,1)

(0,1,1,0) (1,0,0,1) (0,0,0,1)

(1,0,0,1) (0,1,0,1) (0,1,1,0)

(1,0,1,0) (1,1,0,1) (0,0,1,1)

(0,1,1,1) (0,0,1,0) (0,1,0,0)

(1,1,0,1) (1,1,1,1) (1,0,1,0)

(1,0,1,1) (0,1,0,0) (1,1,1,1)

(1,1,1,1) (1,0,1,1) (1,1,0,1)

Table 4. The structure of nonisomorphic semifields of order 16

The number The number of

Semifield IN | of subfields The spectrum elements with equal The opposite

of order 4 of the loop left and right invertible semifield

Vi 2 - {1, 4, 5} 1 Viop ~ Ve

V2 2 1 {1, 3, 4, 5, 6} 3 VT = V2

V3 2 - {1,4, 5, 6} 3 V3op ~ V7

V4 2 1 {1, 3, 4, 5, 6} 3 V40P ~ V5

V8 2 2 {1, 3, 4, 5, 6} 7 ~ v9

Vio 2 1 {1, 3, 5, 6} 3 ViOP = Vio

Vii 2 1 {1, 3, 4, 5, 6} 7 Vt ~ V14

Vi2 2 - {1,4, 5, 6} 1 Vf = V12

Vi3 2 4 {1, 3, 5} 9 VT = V13

Vi5 2 2 {1, 3,4, 5} 7 Vi05p ^ Vie

Vi7 2 1 {1, 3, 4, 5, 6} 3 Vi7 = Vi7

Vi8 2 2 {1, 3, 5, 6} 5 V°8p = Vi8

T24 4 2 {1, 3, 4, 5, 6} 5 rpop rp T 24 — T 24

T25 4 2 {1, 3, 4, 5, 6} 5 TT ^ T50

T35 4 1 {1, 3, 4, 5, 6} 3 ^335^ — T35

T45 4 3 {1, 3, 5} 7 T4°5p — T45

semifield planes is 3. Regular sets of representatives of 5 other isomorphic classes of translation planes are described in [16]. We write them in the following fixed order where O and E are zero and identity matrices, respectively:

O, E,

(2

0

V1 (1

1 0 11 1 0

/ 0 0 0 1 \ 10 10 1111 1001

1

0

00 11

0010 0 0 11

0 1

101

11

0110 1111 0 0 0 1

0 0 1 1

00

1

11

01

/ 0 0 1 1 \ 1001 01 10

11 0 1

01 11

1 \ (

0 11\/1100\

1 1 0 1 , 1 0 0 0

0 1 1 1 , 1 0 0 1

1 1 1 1 0 1 1 1

/110 0010 0110 1010

00 1 \ (

11 11

0100 110 0

100 1

0

1

0

1 \

0 1

1000

011 110/

10

10

0

11

1

1000 110 0

11 1011 0 0 11 0010

1 0 1 1 1 1 0 0 0 1 110 0 0101

/

O, E,

01 11 11

0

0 1 \ 00 10

0 0 0 1 \ 0110 1101 100

10

1111

0010 0 0 11 101 \ 1 1 0

01 111 100 y 0 0 1

0

1

0

0 1

1

10 0 0 0 1 0101 1000

1\/1100\ 1001 0 0 0 1 ) \1 1 1 0

0

,0

,0 1

( 1 1

/ 0 0 1 1 \ 1101 111 011

11

0

0 1

1\ 1

00 10

0 1 \

0010 110 0 0110

100 101 1111 0111

1110 \

0100 1111 0 0 11 101

( 1 0 1 0

0 1\

0

0

0

11

011

1

1111

1000 1011 1001

11

1

1 1 \

1011 0110

0 0 11

O, E,

01

1

0

11

/ 0 0 0 1 \ 1010 1111 1001

01 11

00

0 110

00

1

0

0010 0 0 11

1010

11

10 01 11

00

/

01

011 0 0 1

1 0 1 1 1 1 0 0 1 1 0 1 , 1 0 0 0 0 1 1 1 , 0 0 1 1 1 1 1 1 0 0 1 0

V

( 1 1

/ 0 0 1 1 \ 1001 01 10

11

0 1 1 01

01 11

1 \ (

100 1

11 11

0100 110 0 0 0 0 1 \ 0 0 1 1 /

( 1 0 1 0

0

1 0 1

1111 1110 0111

1000 1 1 1 0 \

0

0

11

1

0 0 0 1 1001 0101

1000 v 1 1 0 0

1111 \

0

0 1

1 1 0 100 010

O, E,

01

1

0

/ 0 0 0 1 \ 1111

V 1

0

110 0 1001

11

111

0010 0 0 11 1010 1101

0110 0 0 0 1 1111 0101

11

00

/

11 1

0 0 1 1 \

V 1

0

11

0 0 1 0

1 0 1 1 1 1 0 0 1 0 0 1 , 1 0 0 0 1 1 1 0 , 1 1 0 1 0 1 1 0 1 0 1 1

1 1 0 100

1111

0100 110 0

0111

100 0 0 1

011

V 1

0 0

1110

1000

10

0

11

/

10

1

1000 110 0

1 1 0 1 1 1 1 0 1 1 , 1 1 0 0 0 1 , 0 1 0 0 1 1 1 0

1 0 1 1 1 1 0 1 , 0 1 0 1 0 1 , 1 0 0 1 1 0 0 1 0 0

O, E,

0 0 0 1 \ 0010 110 0 0101

0010 0 0 11 1010 1101

0 0 1 1 \

0 0 0 1 0100 1011

0100 1111 0111 1001

0

0 1 0 1 \

0 1 1 0 \

0 1 1 1 \

1110

1 0 0 1 \

1 0 1 0 , 1 0 0 0

1 0 1 1 , 1 1 1 1 0 0 1 1

0 1 1 1 1 0 1 0 0 0 1 0

f 10 11 \ f 1100 \ f 110

1 1 0 0 , 1 0 0 1 , 1 0 1 1

1 1 1 0 , 1 1 0 1 , 0 0 0 1

1 0 0 0 0 1 1 0 0 0 1 1

1010 0111

1101

0 1 1 0 1 0 0 0

1 1 1 0 1 1 0 0

1\/1110 \ f 1111 \

0 1 1 0 , 0 1 0 1

0 1 0 1 , 1 0 0 1

1 1 1 1 0 1 0 0

Determining for each of them on the multiplication on formula (1) we get pairwise nonisotopic quasifields Qi, i = 1,2,3,4,5, respectively. For each of them we found the Cayley's table of loop Q* and investigate questions (A)—(C).

The following two Theorems show structure of quasifields Q2 and Q5 which are most similar properties to finite fields.

Theorem 3.1. The quasifield Q2 has a single maximal subfield H and |H| = 4. Each element of Q2 \ H has order 5 and generates a loop Q*.

Unlike Q2, the quasifield Q5 has 3 subfields of order 4:

G1 = {0,e, (0, 0,1,0), (1, 0,1,0)}, G2 = {0,e, (0,1,0,1), (1,1,0,1)},

G3 = {0, e, (0,1,1,1), (1,1,1,1)}

Theorem 3.2. Every maximal subfield of the quasifield Q5 coincides with Gi, G2 or G3 and any element from Q* \ {G1 U G2 U G3} has the order 5 and generates loop Q*. In particular, spectrum of loop Q5 is {1, 3, 5}.

Quasifields Q1, Q3, Q4 has an essential anomalous properties; in particular, any element of each of them is element from some subfield of order 4. One shows

Theorem 3.3. Any semifield Qi, i = 1, 3,4, has 7 maximal subfields of order 4 and their set-theoretic union coincides with Qi. In particular, spectrum of loop Q* is {1, 3}.

Proof. Using the first regular set we define multiplication (1) in quasifield Q1. Further we find Cayley's table of loop Qi (multiplication on the identity element (1,0,0,0) is omitted) (Tab. 5).

Similarly, we find a Cayley's tables for loops Q* and Q*. Any element of each loop Q*, i = 1, 3,4, has identical left and right invertible elements as it show the Cayley's tables. Also each quasifield Qi has the following 7 subfields:

Fi = {0, e, (0,0, 0,1), (1,0,0,1)}, F2 = {0, e, (0, 0,1, 0), (1,0,1, 0)}, F3 = {0, e, (0, 0,1,1), (1, 0,1,1)}, F4 = {0, e, (0,1,0,0), (1,1,0,0)}, F5 = {0, e, (0,1,0,1), (1,1, 0,1)}, F6 = {0, e, (0,1,1,0), (1,1,1,0)}, F7 = {0, e, (0,1,1,1), (1,1,1,1)}.

Clear that the set-theoretic union of 7 different subfields coincides with Qi. In particular, the spectrum of loop Q* is {1,3}. □

Remark 3.1. It was also established that there are quasifields of order 16 with kernel of order 4 having elements of order 3, which are not lie in the any subfield of order 4. For instance, these are Kleinfeld's quasifields S3 and S10.

Table 5. Cayley's table of loop Q*

(0,0,0,1) (0,0,1,0) (0,0,1,1) (0,1,0,0) (0,1,0,1) (0,1,1,0) (0,1,1,1)

(0,0,0,1) (1,0,0,1) (1,1,0,1) (1,1,1,0) (0,1,1,0) (1,0,1,1) (0,0,1,1) (0,1,0,0)

(0,0,1,0) (1,1,1,1) (1,0,1,0) (0,1,0,1) (1,0,1,1) (0,1,0,0) (0,0,0,1) (1,1,1,0)

(0,0,1,1) (0,1,1,0) (0,1,1,1) (1,0,1,1) (1,1,0,1) (1,1,1,1) (0,0,1,0) (1,0,1,0)

(0,1,0,0) (1,0,1,0) (0,0,1,1) (1,0,0,1) (1,1,0,0) (0,1,1,0) (1,1,1,1) (0,1,0,1)

(0,1,0,1) (0,0,1,1) (1,1,1,0) (0,1,1,1) (1,0,1,0) (1,1,0,1) (1,1,0,0) (0,0,0,1)

(0,1,1,0) (0,1,0,1) (1,0,0,1) (1,1,0,0) (0,1,1,1) (0,0,1,0) (1,1,1,0) (1,0,1,1)

(0,1,1,1) (1,1,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1) (1,0,0,1) (1,1,0,1) (1,1,1,1)

(1,0,0,1) (1,0,0,0) (1,1,1,1) (1,1,0,1) (0,0,1,0) (1,1,1,0) (0,1,0,1) (0,0,1,1)

(1,0,1,0) (1,1,1,0) (1,0,0,0) (0,1,1,0) (1,1,1,1) (0,0,0,1) (0,1,1,1) (1,0,0,1)

(1,0,1,1) (0,1,1,1) (0,1,0,1) (1,0,0,0) (1,0,0,1) (1,0,1,0) (0,1,0,0) (1,1,0,1)

(1,1,0,0) (1,0,1,1) (0,0,0,1) (1,0,1,0) (1,0,0,0) (0,0,1,1) (1,0,0,1) (0,0,1,0)

(1,1,0,1) (0,0,1,0) (1,1,0,0) (0,1,0,0) (1,1,1,0) (1,0,0,0) (1,0,1,0) (0,1,1,0)

(1,1,1,0) (0,1,0,0) (1,0,1,1) (1,1,1,1) (0,0,1,1) (0,1,1,1) (1,0,0,0) (1,1,0,0)

(1,1,1,1) (1,1,0,1) (0,1,1,0) (0,0,0,1) (0,1,0,1) (1,1,0,0) (1,0,1,1) (1,0,0,0)

(1,0,0,1) (1,0,1,0) (1,0,1,1) (1,1,0,0) (1,1,0,1) (1,1,1,0) (1,1,1,1)

(0,0,0,1) (1,0,0,0) (1,1,0,0) (1,1,1,1) (0,1,1,1) (1,0,1,0) (0,0,1,0) (0,1,0,1)

(0,0,1,0) (1,1,0,1) (1,0,0,0) (0,1,1,1) (1,0,0,1) (0,1,1,0) (0,0,1,1) (1,1,0,0)

(0,0,1,1) (0,1,0,1) (0,1,0,0) (1,0,0,0) (1,1,1,0) (1,1,0,0) (0,0,0,1) (1,0,0,1)

(0,1,0,0) (1,1,1,0) (0,1,1,1) (1,1,0,1) (1,0,0,0) (0,0,1,0) (1,0,1,1) (0,0,0,1)

(0,1,0,1) (0,1,1,0) (1,0,1,1) (0,0,1,0) (1,1,1,1) (1,0,0,0) (1,0,0,1) (0,1,0,0)

(0,1,1,0) (0,0,1,1) (1,1,1,1) (1,0,1,0) (0,0,0,1) (0,1,0,0) (1,0,0,0) (1,1,0,1)

(0,1,1,1) (1,0,1,1) (0,0,1,1) (0,1,0,1) (0,1,1,0) (1,1,1,0) (1,0,1,0) (1,0,0,0)

(1,0,0,1) (0,0,0,1) (0,1,1,0) (0,1,0,0) (1,0,1,1) (0,1,1,1) (1,1,0,0) (1,0,1,0)

(1,0,1,0) (0,1,0,0) (0,0,1,0) (1,1,0,0) (0,1,0,1) (1,0,1,1) (1,1,0,1) (0,0,1,1)

(1,0,1,1) (1,1,0,1) (1,1,1,0) (0,0,1,1) (0,0,1,0) (0,0,0,1) (1,1,1,1) (0,1,1,0)

(1,1,0,0) (0,1,1,1) (1,1,0,1) (0,1,1,0) (0,1,0,0) (1,1,1,1) (0,1,0,1) (1,1,1,0)

(1,1,0,1) (1,1,1,1) (0,0,0,1) (1,0,0,1) (0,0,1,1) (0,1,0,1) (0,1,1,1) (1,0,1,1)

(1,1,1,0) (1,0,1,0) (0,1,0,1) (0,0,0,1) (1,1,0,1) (1,0,0,1) (0,1,1,0) (0,0,1,0)

(1,1,1,1) (0,0,1,0) (1,0,0,1) (1,1,1,0) (1,0,1,0) (0,0,1,1) (0,1,0,0) (0,1,1,1)

4. Semifields of order 32

In 2011 all translation planes of order 32 and their regular sets were described by U. Dempwolff and R.Rockenfeller [16,22]. (See also R.J.Walker [23].) There are 9 of these planes up to isomorphisms including 5 semifield planes and a Desargues plane. A coordinatizing set here is a 5-dimentional space W over the field Z2.

Regular sets of non-Desargues semifield planes of order 32 are described in [16]. We denote their by Ri (1 < i < 5), according to [17]. Let Pi (1 < i < 5) be a semifield, which corresponding regular set Ri. In particular, using the regular set

( x

z

z + s z + w + s \ y + z + w

y

x + z + w w z + s w + s

z

y + w

x

s y

w w + s y + w x + z + w z

s w z + w y + z + s x + z

\

and multiplication (1) we obtain the semifield P5. The following theorems are proved in [17]. Theorem 4.1. The minimal subfield Z2e in every semifield Pi, i = 1, 2, 3,4, is maximal and

each element of order > 1 generates the loop P*. The spectrum of loop P* is {1, 4, 5, 6, 7} for i = 1, 2; it coincides with {1, 4, 5, 6, 7, 8} for i = 3 and with {1, 5, 6, 7, 8, 9} for i = 4.

Theorem 4.2. The semifield P5 has a single subfield H and \H| = 4. The spectrum of the loop P| is {1, 3,4, 5, 6, 7, 8}. Each element of P5 \ H has order > 3 and generates the loop Pi*.

Remark 4.1. Theorem 5.2 distinguishes a semifield of order 32 having anomalous property of subfields (compared with finite fields): the semifield P5 of order 25 has a subfield of order 22. The similar subfield S is constructed by Rua [6, Corollary 1]. In connection with the Wene's hypothesis he shows that the loop S* is not right primitive. However, it is possible to prove that this loop S* is one generated.

The problem on the structure of semifields of order 32 is more difficult than the case of semifields of order 16 which studied in Section 2. Up to isomorphism, there exist 2502 semifields of order 32 and they form 6 isotopic classes corresponding to six of pairswise nonisomorphic planes n(i) (0 < i < 5) [6]. This shows the following table from [6] (Tab. 6).

Table 6. Isomorphic classes of semifields of order 32

Plane n(0) n(1) n(2) n(3) n(4) n(5)

Left and right primitive 1 961 961 180 186 186

Only left primitive 0 0 0 6 0 7

Only right primitive 0 0 0 6 7 0

Neither L. nor R. prim. 0 0 0 1 0 0

This work was supported by the Russian Foundation Basic Research (project 12-01-00968).

References

[1] A.G.Kurosh, Lectures on general algebra, New-York, Chelsea Publishing, 1965.

[2] D.R.Hughes, F.C.Piper, Projective planes, Springer-Verlag, New-York Inc, 1973.

[3] H.Liineburg, Translation planes, Springer-Verlag, Berlin-New-York, 1980.

[4] N.L.Johnson, V.Jha, M.Biliotti, Handbook of finite translation planes, London-New-York, 2007.

[5] G.P.Wene, On the multiplicative structure of finite division rings, Aequationes Math., 41(1991), 791-803.

[6] I.F.RUa, Primitive and non primitive finite semifields, Commun. Algebra, 32(2004), no.2, 793-803.

[7] N.D.Podufalov, About functions on linear spaces, which are connected with finite projective planes, J. Algebra and Logic, 41(2002), no. 1, 83-103.

[8] The Kourovka Notebook (unsolved problems in group theory), 15-th ed., Novosibirsk, Inst. of Math., SB RAN, 1992.

[9] L.E.Dickson, Linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc, 7(1906), 370-390.

10] J.R.Wesson, On Veblen-Wedderburn Systems, Amer. Math. Monthly, 64(1957), no. 9, 631-635.

11] E.Kleinfeld, Techniques for enumerating Veblen-Wedderburn systems, J. Assoc. Comput. Mach., 7(1960), 330-337.

12] D.E.Knuth, Finite semifields and projective planes, J. Algebra, 2(1965), 182-217.

13] U.Dempwolff, A.Reifart, The Classification of the translation planes of order 16, Part I, Geom. Dedicata, 15(1983), 137-153.

14] N.L.Johnson, T.G.Ostrom, Tangentially transitive planes of order 16, J. Geometry, 10(1977), no. 2, 146-163.

15] N.L.Johnson, Elations in translation planes of order 16, J. Geometry, 20(1983), 101-110.

16] U.Dempwolff, File of Translation Planes of Small Order, www.mathematik.um — kl.de/ ~ dempw/dempw_ Plane.html.

17] P.K.Shtukkert, Quasifields and projective translation planes of even small orders, Izvestiya Irkut. Gos. Univ., 7 (2014), no. 1, 144-159 (in Russian).

18] M.Hall, Theory of groups, Macmillan, 1959.

19] T.Oyama, On quasifields, Osaka J. Math., 22(1985), no. 1, 35-54.

20] A.I.Mal'cev, Foundations of linear algebra, W.H. Freeman, 1956.

21] O.V.Kravcova, P.K.Kurshakova (Shtukkert), On the isomorphism of semifield planes, Vest-nik Krasnoyar. Gos. Technic. Univ., 42(2006), 13-19 (in Russian).

22] R.Rockenfeller, Translationsebenen der Ordnung 32, Diploma Thesis, FB Mathematik, University of Kaiserslautern, 2011.

23] R.J.Walker, Determination of division algebras with 32 elements, Proc. Symp. Appl. Math. XV, Amer. Math. Soc, 1962, 83-85.

Вопросы строения квазиполей порядка 16 и 32

Владимир М. Левчук, Полина К. Штуккерт

Известный метод построения конечных проективных плоскостей трансляций (аналогично, полуполевых плоскостей) основан на их соответствии с квазиполями (соответственно, с полуполями) того же порядка. В статье 'рассматриваются вопросы структурного строения конечного квазиполя (возможные максимальные подполя, свойства цикличности мультипликативной лупы ненулевых элементов и возможные порядки элементов). Для конечных квазиполей малых четных порядков найдены аномальные свойства.

Ключевые слова: проективная плоскость трансляций, квазиполе, полуполе, мультипликативная лупа, порядки элементов.