CC BY
1
Journal of Siberian Federal University. Mathematics & Physics 2023, 16(6), 705—719
EDN: ARCPOE
УДК 519.145
Linear Autotopism Subgroups of Semifield Projective Planes
Olga V. Kravtsova*
Daria S. Skok
Siberian Federal University
Krasnoyarsk, Russian Federation
Received 10.03.2023, received in revised form 15.06.2023, accepted 04.09.2023
Abstract. We investigate the well-known hypothesis of D. R. Hughes that the full collineation group of
non-Desarguesian semifield projective plane of a finite order is solvable (the question 11.76 in Kourovka
notebook was written down by N. D. Podufalov). This hypothesis is reduced to autotopism group that
consists of collineations fixing a triangle. We describe the elements of order 4 and dihedral or quaternion
subgroups of order 8 in the linear autotopism group when the semifield plane is of rank 2 over its kernel.
The main results can be used as technical for the further studies of the subgroups of even order in
an autotopism group for a finite non-Desarguesian semifield plane. The results obtained are useful to
investigate the semifield planes with the autotopism subgroups from J. G. Thompson’s list of minimal
simple groups.
Keywords: semifield plane, autotopism, homology, Baer involution, Hughes’ problem.
Citation: O.V. Kravtsova, D.S. Skok, Linear Autotopism Subgroups of Semifield
Projective Planes, J. Sib. Fed. Univ. Math. Phys., 2023, 16(6), 705-719. EDN: ARCPOE.
Introduction
It is well-known that the geometric properties of projective plane are closely connected with
the algebraic properties of its coordinatizing set. So, a finite Desarguesian projective plane is
coordinatized by the field, a translation plane by the quasifield.
The study of finite semifields and semifield planes started ago with the first examples con-
structed by L. E. Dickson in 1906. A semifield is called a non-associative ring Q = (Q, +, ■) with
identity where the equations ax = b and ya = b are uniquely solved for any a,b G Q, a = 0.
The abcense of an associative law in a semifield leads to a number of anomalous properties in
comparison with a field or a skewfield or even a near-field.
By the mid-1950s, some classes of finite semifield planes had been found. All of them had the
common property that the collineation group (automorphism group) is solvable. So D. R. Hughes
conjectured in 1959 that any finite projective plane coordinatized by a non-associative semifield
has the solvable collineation group. This hypothesis is presented in the monography [1]; it is
proved also that the hypothesis is reduced to the solvability of an autotopism group as a group
fixing a triangle. Moreover, the hypothesis is reduced to linear autotopism subgroup over the
kernel. The Hughes' problem attracted the interest of a wide range of researchers who proved
the collineation group solvability for an extensive list of semifield planes with certain restrictions.
*ol71@bk.ru https://orcid.org/0000-0002-6005-2393
^skokdarya@yandex.ru
© Siberian Federal University. All rights reserved
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
In 1990 the problem was written down by N. D. Podufalov in the Kourovka notebook ( [2], the
question 11.76).
We represent the approach to study Hughes’ problem based on the classification of finite
simple groups and theorem of J. G. Thompson on minimal simple groups. The spread set method
allows us to identify the conditions when the semifield plane with certain autotopism subgroup
exists. This method can be used also to construct examples, including computer calculations.
The elimination of some groups from Thompson’s list as autotopism subgroups allows us make
progress in solving the problem.
We consider, mostly, the case when a semifield plane has a rank 2 over its kernel basing on
the theory of M. Biliotti and co-authors [3]. Nevertheless, some results are generalised to N -rank
case.
It is shown by the first author in [4, 5], that an autotopism of order two has the matrix
representation convenient for calculations and reasoning. Here we use the spread set method
to describe the geometric sense of an autotopism of order 4. The matrix representation of
this autotopism allows us to prove the criterion of existence for the dihedral and quaternion
subgroups of order 8 in an autotopism group. We present the examples of semifield planes of
minimal order 625 with this property.
1. Main definitions and preliminary discussion
We use main definitions, according [1,6], see also [7-9], for notifications.
Consider a linear space Q, n-dimensional over the finite field GF(ps) (p is prime) and the
subset of linear transformations R C GLn(ps) U {0} such that:
1) R consists of pns square (n x n)-matrices over GF(ps);
2) R contains the zero matrix 0 and the identity matrix E;
3) for any A, B G R, A = B, the difference A — B is a nonsingular matrix.
The set R is called a spread set [1]. Consider a bijective mapping в from Q onto R and
present the spread set as R = {в(у) | y G Q}. Determine the multiplication on Q by the rule
x * у = x ■ в(у) (x,y G Q). Then (Q, +, *) is a right quasifield of order pns [6,10]. Moreover, if
R is closed under addition then (Q, +, *) is a semifield.
Note, that if we use a prime field Zp as a basic field then the mapping в is presented using
only linear functions; it greatly simplifies reasoning and calculations (also computer).
A semifield Q coordinatizes the projective plane n of order |n| = |Q| such that:
1) the affine points are the elements (x,y) of the space Q ® Q;
2) the affine lines are the cosets to subgroups
V(ж) = {(0, у) | у G Q}, V(m) = {(x,xe(m)) | x G Q} (m G Q);
3) the set of all cosets to the subgroup is the singular point;
4) the set of all singular points is the singular line;
5) the incidence is set-theoretical.
The solvability of a collineation group Aut n for a semifield plane is reduced [1] to the solv-
ability of an autotopism group Л (collineations fixing a triangle). Without loss of generality, we
can assume that linear autotopisms are determined by linear transformations of the space Q ® Q:
A : (x,y) ^ (x,y)
A0 B0
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
here the matrices A and B satisfy the condition (for instance, see [11])
A-1e(m)B € R V6(m) € R. (1)
The collineations fixing a closed configuration have special properties. It is well-known [1],
that any involutory collineation is a central collineation or a Baer collineation.
A collineation of a projective plane is called central, or perspectivity, if it fixes a line pointwise
(the axis) and a point linewise (the center). If the center is incident to the axis then a collineation
is called an elation, and a homology in another case. The order of any elation is a factor of the
order |^j of a projective plane, and the order of any homology is a factor of |n| — 1. All the
perspectivities in an autotopism group are homologies and form the cyclic subgroups [12]:
Hi
M E) Iм e r4 h= { (0«)
Нз
{(M M |м e R}
м e r*
}■
The matrix subsets Ri, Rm, Rr are defined by a spread set [12]:
Ri = {M € GLn(ps) U {0} | MT = TM VT € R},
Rm = {M € R | MT e R VT € R},
Rr = {M € R | TM € R VT € R},
they are called left, middle and right nuclei of the plane n respectively. These subfields in
GLn(ps) U {0} are isomorphic to correspondent nuclei of the coordinatizing semifield Q:
Nl = {x € Q | (x * a) * b = x * (a * b) Va, b € Q},
Nm = {x € Q | (a * x) * b = a * (x * b) Va, b € Q},
Nr = {x € Q | (a * b) * x = a * (b * x) Va, b € Q}.
The plane n is Desarguesian (classic) iff Q is a field, then R ~ Q ~ GF(pns).
An autotopis group of a semifield plane of odd order contains three involutory homologies:
"1 = — E) € H1, h2 = (0 -e) € H" h3 = h'h'2 = {—0E —e)
e H3
A collineation of a projective plane n of order m is called Baer collineation if it fixes pointwise
a subplane of order = ^/m (Baer subplane). We use the results on the matrix representation
of a Baer involution т € Л and of a spread set obtained by M. Biliotti with co-authors [3] and by
the first author in [4,5].
2. Linear autotopisms of order 4
We consider now the case when a semifield plane n has a rank 2 over its kernel, |n| = |^ |2.
To simplify the notification we use K = Nl ~ GF(q), q = pn. The point set of the plane is
П = {(xi,x2,yi,V2) | xi,yi € GF(q)},
- 707-
Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
the spread set R consists of (2 x 2)-matrices determined its second row:
r = { e(v,u)= f vu) 9(vuu))\v,u e GF(,)} .
Here the functions f and 9 are additive:
f (vi,Ul ) + f (V2,U2) = f (Vl + V2 ,Ul + U2 ),
g(vi, ui) + g(v2,U2) = g(vi + V2,ui + U2), vi, V2,ui, u2 e GF(q),
so f and 9 are the additive polynomials:
n-i n-i
f (v, u) = Y,(fjup + Fjvp), 9(v, u) = Y,(9iup + Gjvp), f ,Fj, 9j,Gj e GF(q).
j=0 j=0
The autotopism group Л consists of semi-linear transformations of the linear space:
A :(x,y) ^ (xa ,Va^^ B) ’
where a is a basic field automorphism:
xa = (xi,X2)a = (xp ,Xp ).
Evidently, that the subgroup Л0 of linear autotopisms (t = 0) is normal in Л and the factor Л/Л0
is isomorphic to a subgroup of Aut K. Therefore, the solvability problem is reduced to the linear
autotopism subgroup Л0.
G. E. Moorhouse in 1989 proved [13]:
Lemma 1. Let n be a projective plane of order n2, n = 2 or 3 (mod 4), and G is a cyclic
collineation group of order 4. Then the involution in G is central.
We will expand Moorhouse’ result for |n|
p2n if p = —1 (mod 4).
Let n be a non-Desarguesian semifield plane of order q2 with the kernel K ~ GF(q) (q = 2n).
If т e Л0 is an involution then it is Baer, and we can propose that, in appropriate base, it has a
Jordan normal form (see [3]):
(1 1 0 0
0 1 0 0
0 0 1 1
0 0 0 1
C 01- L-(;!)
(2)
The spread set R e GL2(q) U {0} consists of matrices
d(v, u
)=
v + u + m
v
(v)
f (v) + m(u)
u
,
v,u e GF(2n).
(3)
Lemma 2. The linear autotopism group Л0 of a semifield projective plane n of order 22n with
the kernel K ~ GF(2n) does not contain elements of order 4.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
Proof. Let a G Л0 be an autotopism of order 4, a4 = e. Then a2 = т is a Baer involution (2),
because hi,h2,h3 G Л for p = 2. Let
e в
then A2 = B2 = L, AL = LA, BL = LB. So we have
a=с»:). в=$ $.
where
(
«I = i,
b2 = 1,
«102 + 0202 = 1; l bib2 + b2b2 = 1.
The systems have no solution in a field of the characteristic 2, the lemma is proved. □
Theorem 2.1. Let n be a semifield non-Desarguesian plane of order 22n with the kernel K ~
GF(2n). Then the Sylow 2-subgroup of the linear autotopism group Л0 has an order at most 2.
Proof. From the lemma, the Sylow 2-subgroup S С Л0 is elementary Abelian. Let т,а G S,
where т is (2). Then
1 a 0 0
0 1 0 0
0 0 1 b
0 0 0 1
AB.
Consider the condition (1)
A-20(v, u)B G R Vv, u G GF(2n)
for the spread set (3). For 0(0,1) = E we have
A 'LB = (1 a)" (1 b) = (1 b + “) G R b = a.
Further, for 0(v, 0):
and
/1 a\ f v + m(v) f (v)\ (1 a\
V0 VV v о )\o 1) =
(v + m(v) + av f (v) + av + m(v)a + a2v\ . .
= 0(v,va),
v va
m(va) = av + m(v)a + a2v, Vv G GF(2n).
Consider the polynomial m(v):
✓4 24 2n—1
m(v) = mo v + miv + m2v + ... + mn-iv ,
О 0 A A Q A 0 0
mova + miv2a2 + m2v a + ... = mova + miv2a + m2v a + ... + va + va2, a + a2 =0.
If a = 0 then a = e; if a =1 then a = т. The theorem is proved.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
Let now p > 2, |n| = p2n, K ~ GF(pn). We will not consider the case p = — 1 (mod 4): this
case is more complicated for a semifield plane of arbitrary rank, see [14]. If p = 1 (mod 4) then
the prime field Zp of K contains an element i such that i2 = —1. We have iE = E + • • • + E,
therefore iE e Rr П Rm and the linear autotopism group Л0 contains the homologies of order 4:
ai
(iE 0 \
\0 e)
e Hi,
a2
(0 iE)e H2, aia2 e
0 iE
As has be proven in [4], a Baer involution т e Л0 can be written as
—1 0 0 0
0 1 0 0
0 0 —1 0
0 0 0 1
(Lу ‘=to1;»
(4)
(5)
for appropriate Jordan base. The spread set R consists of matrices
0(v,u)= (m(u) f (v)) , v,u e GF(pn),
\ v u )
where the functions m and f are injective additive polynomials, m(1) = 1, f (1) = ±1.
The following theorem expands Moorhouse' lemma 1.
Theorem 2.2. Let n be a non-Desarguesian semifield plane of order p2n with the kernel K ~
GF(pn), p is prime, p = 1 (mod 4), and a e Л0 is a linear autotopism of order f. Then a2 is a
homology and either a e (a1, a2) or n admits a linear Baer involution т and a e (a1, a2, т).
Proof. Let a is not homology,
(A B)
then a2 is the homology h1, h2, h3 or a Baer involution т (4).
If a2 = т then from A2 = B2 = L, AL = LA, BL = LB we have
A=Co A B=Co- 0.
where a1,b1 e {i, —i}, a2, b2 e {1, —1}. Thus, the autotopism a can be represented as a product
±i 0 0 0
0 ±1 0 0
0 0 ±i 0
0 0 0 ±1
(L 0) (M 0 ) (-E 0)
Vo l) V0 m)\ 0 e) =т^ 1
or a = т^2 or a = трЕз, where p =
M0
0M
for the collineation p for any matrix (5): M-19(v,u)M e R. For any v e K and u = 0 we have:
M = 0i 10 . We consider the condition (1)
M-19(v, 0)M
—i 0 0 f(v) i 0 = 0 —if(v)
V0 1) Vv 0 J V0 1) Viv 0 )
R,
f (iv) = —if (v). It contradicts with the additivity of f (x), because i e Zp and so f (iv) = if (v).
Therefore, the case a2 = т is impossible, and a2 is a homology.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
Let a2 = hi. Then A2 = —E, B2 = E, and the following Jordan normal form are possible:
±iE, ±iL for A and ±E, ±L for B. If A = ±iE or B = ±E then a £ (ai, a2). The remaining
possibility
“ = d LPf A
= т is the Baer involution, a £ (ai, a2, т).
For a2 = h2 and a2 = h3 we obtain the analogous result. The theorem is proved. □
Note that the homologies generate the normal subgroup in the autotopism group [1]. There-
fore, if F < Л0 is a simple non-Abelian subgroup then it does not contain elements of order 4 for
p ф — 1 (mod 4). Further, any elementary Abelian subgroup of Л0 is of order at most 8. Maxi-
mal its order is for (т, hi, h2); but (hi, h2) is normal in Л0, thus |F| is either odd or 2 • (2m + 1).
This contradicts to conjecture that F is simple non-Abelian group.
Corollary 1. Let n be a non-Desarguesian semifield plane of order p2n with the kernel K ф
GF(pn), p is prime, p ф 1 (mod 4). Then its autotopism group contains no simple non-Abelian
subgroups.
For p = 2 the more significant result has been proven by M. J. Ganley in 1974 ( [15], see
also [16]).
Theorem 2.3. Let Q be a finite semifield of order 2s. If Q has dimension 2 over one of its
nuclei then its autotopism group is solvable.
Extend the results obtained by the information on coordinatizing semifield automorphisms.
It has been proven in [11], that the linear transformation x ^ xA is the automorphism of a
semifield Q iff the matrix
(A A) (6)
is the autotopism of the semifield plane n with the condition A-i9(m)A = 9(mA) for any m £ Q.
Therefore, we have the following result.
Corollary 2. Let Q be a non-associative semifield of order p2n with the left nucleus K ф GF(pn),
p is prime, p ф —1 (mod 4). Then the automorphism subgroup AutKQ of Q that fixes K has a
Sylow 2-subgroup of order at most 2.
Proof. For the even case p = 2 the result is a direct consequence of the theorem 2.1. Let p ф 1
(mod 4) and the transformation x ^ xA be an automorphism from AutK Q. Then we can assume
that A £ GL2(pn). According to [11], the involution (6) may be Baer only. Up to base chosen,
we can suppose A = L. The centralizer of т in Л0 contains involutions т, hiT, h2T, h3T only. All
these possibilities lead to the contradiction, because the involution т • (Ь^т) is not Baer. Thus,
the elementary Abelian 2-subgroup of AutKQ is of order at most 2. Finally, if the autotopism
a (6) is of order 4 then a2 is an involutory homology h3 with A = —E; we have the contradiction.
□
leads to a
(aihkfi hk22)-
L L
1
3. Dihedral and quaternion subgroups
The question on autotopism subgroups isomorphic to D8 or Q8 is explained by the fact that
such subgroups are contained in the Sylow 2-subgroup of a large number of simple non-Abelian
groups. For semifield plane of arbitrary rank over the kernel, the first author proved [8] that a
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
dihedral autotopism subgroup of order 8 must contain the homology if p = 1 (mod 4). Now we
describe the matrix representation of subgroup
H = (а, в | a4 = в2 = 1, fiafi = a 4) ~ D8
(7)
—i 0 0 0 0 1 0 0
a = 0 0 i 0 0 —i 0 0 • h\1 hk2, в= 1 0 0 0 0 0 0 1
0 0 0 i 0 0 1 0
and the spread set matrices. The generalization of this result for N -dimensional case see in the
next section.
Theorem 3.4. Let n be a non-Desarguesian semifield plane of order p2n with the krenel K ~
GF(pn), p is prime, p = 1 (mod 4), and the linear autotopism group Л0 contains a subgroup
isomorphic to the dihedral group of order 8 (7). Then the base of 4-dimensional vector space
over K can be chosen such that H = (а, в),
— 1, i £ Zp,
k1, k2 £ {0,1}. The spread set of n consist of matrices (5), where m and f are injective involutory
functions on GF(pn), m(m(x)) = x, f (f (x)) = x.
Proof. If a £ (a1,a2) then a £ Z(Л) and the condition вав = a-1 is not satisfied for any
autotopism в. Therefore, the plane n admits the Baer involution т, it has the spread set (5),
and a £ (a1 ,a2,r), from the theorem 2.2. Then a2 is one of involutory homologies h1, h2, h3.
Consider the possible cases.
Let a2 = h1. Then, up to involutory homologies, a = ^ = a1T, and в(a1T)в =
о D1 'then
(a1T)3в, втв = h1T. Denote в = D) ’ then C2 = D2 = E, CLC = —L, DLD = L, and
either D = ±E or D = ±L. If D = ±E then ^
is the involutory homology, and so
C0
0E
C = —E, this contradicts to condition. Therefore, up to involutory homology, D = L,
в (C 0) (CL 0)
в =(о l) = (o e) •T’
and вт is the homology of order 4: вт = a1 or вт = a3, CL = ±iE, C = ±iL, C2 = —E = E.
Thus, the case a2 = h1 is impossible.
By similar reasoning we come to a contradiction in the case a2 = h2.
Let now a2 = h3, and a =
iL 0
0 iL
= a1a2T, without the involutory homologies. Then
в^^2t)в = (a1a2T)3 = a.1a2h3T, втв = h3T, CLC = DLD = —L, and we have
c=(;. о)- с=(; 10)
Consider the transition matrix T:
T =
1000
0 c 0 0
0 0 10
Vo 0 0 d)
(8)
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
Then, for the new base,
TaT 1 = a,
твт-1
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
To complete the proof, it is enough to test the involutority of the functions m and f. Indeed,
в is a collineation, and, for any matrix 0(v, u) from the spread set R, the product
(0 1) (m(u) f (v)) (0 1) = ( u v )
\1 оД v u ) \1 0J \f(v) m(u)J
must belong to R, see (1). Therefore, m(m(u)) = u and f (f (v)) = v for all v,u G GF(pn). The
theorem is proved. □
Now let the linear autotopism group Л0 contains a quaternion subgroup of order 8:
F = (a, 7 | a4 = 74 = 1, a2 = 71, arfa = 7) ^ Qg.
(9)
Theorem 3.5. Let n be a non-Desarguesian semifield plane of order p2n with the kernel K ~
GF(pn), p is prime, p = 1 (mod 4), and the linear autotopism group Л0 contains a subgroup
isomorphic to the quaternion group of order 8 (9). Then the base of 4-dimensional vector space
over K can be chosen such that F = (a, 7),
-i 0 0 0 0 1 0 0
a = 0 0 i 0 0 -i 0 0 • hi1 hk22, 7= -1 0 0 0 0 0 0 1
0 0 0 i 0 0 -1 0
i --- -1, i G Zp,
ki, k2 G {0,1}. The spread set of n consist of matrices (5), where m and f are injective involutory
functions on GF(pn), m(m(x)) = x, f (f (x)) = x.
Proof. If either a or 7 belongs to subgroup (a1,a2) < Z(Л0) then a.7 = 7a, it is impossible.
Therefore, the plane n admits a Baer involution т (4) and, for instance, a G (a.1,a2,r). Evident,
that we can ignore the involutory homologies factors, because (a, 7) ~ Q8 leads to (ah^1 h!j2,7) ~
Qg. So, we suppose further a = a1r, a = a2r or a = a1a2r.
In the first case a2 = 72 = h1. Notify
7=(c D)’ C=-E d2=E
and consider the condition a7a = 7:
(a1r )7(a1r ) = 7, F1T7T = 7, -LCL = C, LDL = D.
Then D G {E, -E,L, -L}, but the case D
±E is impossible:
C
0
order 4, C = ±iE, a.7 = 7a. Therefore, up to involutory homologies, D
0
E
L,
is a homology of
7
C 0 CL 0
V0 l) v 0 e) •r
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
then 7т is the involutory homology too (from CLCL = E), that is 7т = hi, 7 = h\r, and
a.7 = 7a. This contradiction shows that the case a2 = hi is impossible; for a2 = h2, similarly.
Let now a2 = y2 = h3, a = ,° ) = aia2T. Then
0 iLj
a^a = (aia2T )y (aia2T) = hih-2 T7T = h^TjT = 7,
and we have the conditions -LCL = C, -LDL = D, C2 = D2 = —E, leading to
C = (A 0) • D = (Si 0) .
Choose the transition matrix T (8), then for the new base we obtain
TaT -
T7T -
0 1 0 0
—1 0 0 0
0 0 0 1
0 0 —1 0
Prove the involutority of m and f form the condition (1):
(—10)1 ^ (—10) = f ,—ы)£ R v’-'‘e GF(pn)-
Thus, f (—f (v)) = —v, m(m(u)) = u, and the additivity leads to f (f (v)) = v. The theorem is
proved. □
i
i
a
Remark 1. Note that, according to the theorem 2.2 on autotopisms of order 4, the collineation
Y must be the product of homologies to a Baer involution. Indeed,
0 i 0 0
—i 0 0 0
7 = aia2a, a = 0 0 0 i
0 0 —i 0
where a is the Baer involution fixing pointwise the subplane
= {(^i, ixi, yi,iyi) | xi,yi e GF(pn)}.
Rewrite the autotopism 7 of order 4 as
Y
0 1 0 0 0 1 0 0 —1 0 0 0
—1 0 0 0 1 0 0 0 0 1 0 0 = вт.
0 0 0 1 0 0 0 1 0 0 —1 0
0 0 —1 0 0 0 1 0 0 0 0 1
We see that (a, в) — D8 and (a,@T) ~ Q8, and the following corollary is proved.
Corollary 3. Let n be a non-Desarguesian semifield plane of order p2n with the krenel K —
GF(pn), p is prime, pj = 1 (mod 4). The linear autotopism group Л0 contains a subgroup
isomorphic to the quaternion group of order 8 iff Л0 contains a subgroup isomorphic to the
dihedral group of order 8.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
4. Examples
Construct the semifield planes of minimal order satisfying the condition of theorems 3.4 and
3.5. It is well-known [1] that a semifield of order p2 is a field GF(p2). Therefore the minimal
examples are the planes of order 54 = 625. Let the field K ~ GF(25) is an algebraic extension of
Z5, K = Z5(a), where a is the root of the irreducible polynomial x2 + 3x + 3 G Z5[x]. Then the
semifield Q of order 625 is the vector space Q = {x = (v,u) | v,u G K} with the multiplication
law y * x = y ■ 0(x) (x, y G Q); the spread set R consists of matrices (5). The functions m and f
are the polynomials from K [x]:
m(u) = mou + myu5, ф(v) = fov + fiv5,
which satisfy the conditions m(m(u)) = u, f (f (v)) = v for all u,v G K, m(1) = 1, f (1) = ±1.
There exist 34 pairs of functions m, f such that det 9(v, u) = 0 only for (v, u) = (0,0). Therefore,
we obtain 34 semifield planes of order 625 with the kernel of order 25 which admit the linear
autotopism subgroup isomorphic to D8 (or Q8). At most 11 pairwise non-isomorphic planes are
among them. The isomorphism is either multiplication by a suitable matrix (i.e. changing of
base) or the automorphism of K:
(mo, mi, fo, fi) ^ (m^ ml,f0 ,fl)
The table below represents the coefficients m;, fi together with the nuclei of the semifields.
Table 1. Information on the planes of order 625
№ mo mi fo fi Ni Nm = Nr
1 0 1 0 a K {(0,y) 1 y G K}
2 0 1 0 a +1 INml = INr 1 = 25
3 0 1 2a + 1 2
4 0 1 2a + 1 3
5 4a + 2 a + 4 a + 3 a + 2 {(0,y) 1 y G Z5}
6 4a + 2 a + 4 a + 3 3a + 4 K l I СЛ
7 3a + 4 2a + 2 0 a +1
8 3a + 4 2a + 2 2a + 1 2
9 3a + 4 2a + 2 2a + 1 2a
10 3a + 4 2a + 2 2a + 1 2a + 2 {(x,y) 1
11 a + 3 4a + 3 a + 3 a + 2 K x G {0, a + 3, 2a + 1, 3a + 4, 4a + 2}, y G Z5}
INmI = INr I = 25
5. Generalization for arbitrary dimension
Here we consider the case when a semifield plane n has the order pN, without restriction to
the order of the kernel. In this case we can represent the point set of n as a 2N-dimensional
vector space over Zp, with the spread set R C GLN(p) U {0}. Some results from the previous
sections can be generalized for any N and p = 1 (mod 4).
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Linear Autotopism Subgroups of Semifield Projective Planes
Let n be a non-Desarguesian semifield plane of order pN (p > 2 be prime). According to the
results of [5], if the autotopism group Л contains the Baer involution т then N = 2n is even and
we can choose the base of 4n-dimensional linear space over Zp such that
т = (L L) ' L = (f E)■ (10)
The spread set R in GL2n(p) U {0} consists of matrices
wu)=m^)), ас
where V £ Q, U £ K, Q,K are the spread sets in GLn(p) U {0}, K is the spread set of the Baer
subplane nT, m, f are additive injective functions from K and Q into GLn(p) U {0}, m(E) = E.
Note that throughout the section, the blocks-submatrices have the same dimension by default.
Instead of linear autotopism group Л0 we will consider the autotopism group Л:
л={ а в
A, B £ GLn(p), A-1d(m)B £ R V0(m) £ R
■
Unfortunately, we can not now extend the result of the theorem 2.2 to the general case. The
geometric sense of order 4 autotopism a was presented in [14] when a2 = т is a Baer involution.
Perhaps, one will construct the examples illustrating the matrix representation of a spread set
in this case; there is no evident contradiction.
Theorem 5.6. Let n be a non-Desarguesian semifield plane of order pN, p is prime, p = 1
(mod 4), and a £ Л is an autotopism of order f. If a2 is a homology then either a £ (ai, a2) or
n admits a Baer involution т and a £ (a1, a2,T).
where A2 = -E, B2 = E. Therefore
A = diag (i, -i) = ±iE, B = diag (1, -1) = ±E. The number of 1 among the diagonal elements
of B equals to the number of -1, because else we have the autotopism that fixes more than a Baer
subplane, it is impossible. So, we can assume, up to base changing and involution homologies
factors, that B = L and A = iL, a = a1т. For a2 = h2 and a2 = h3 the consideration is similar.
□
We extend now the main result of [8]:
Theorem 5.7. Any non-Desarguesian semifield plane n of order pN, where p > 2 is prime and
p = 1 (mod 4), does not admit an autotopism subgroup isomorphic to the dihedral group of order
8 without homologies.
Proof. Let a2 = h1 and a £ (a1,a2). Then a
A0
0B
Denote the following autotopisms:
-iE0 0 0 ^
a = 0 iE 0 0 hk1 hk22,
0 0 -iE 0
\ 0 0 0 iE (12)
0 ^ 0 E 0 0 ^
0E0
в = E00 0 -E 0 0 0
0 0 0 E , Y 0 0 0 E ,
\0 0 E 0 0 0 -E 0
where i £ Zp, i2 = -1, k1, k2 £ {0, 1}. Using the result of [18], we prove.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
Theorem 5.8. Let n be a non-Desarguesian semifield plane of order pN, p is prime, p = 1
(mod 4). The autotopism group Л contains a subgroup H ^ D8 (7) iff it contains a subgroup
F ~ Q8 (9). Then N = 2n ^ 4, the subgroups H and F contains the involutory homology h3,
the plane n admits a Baer involution т (10). The base of linear space can be chosen such that
H = (а, в), F = (a, 7), where the autotopisms are (12). The spread set R consists of matrices
(11), where V € Q, U € K, the sets Q,K C GLn(p) U {0} are closed under addition. The
additive injections m : K ^ K and f : Q ^ Q are non-trivial involutions.
Proof. The result for Q8 had been proven in [18]. We will repeat now the proof of the theorem 3.4
with generalization for N -dimensional case. Let а € H ^ D8, a4 = e. Then a2 is the homology
by the theorem 5.7.
1. If a2 = hi then a € (ал,a2) C Z(Л). Therefore we can assume, up to base changing, that
a = а1т, the spread set R consists of matrices (11). Further, let the Baer involution в be the
matrix в
C 0
0D
Then, from ав = ва 1 we have
C-(C C;) ■ D=(D1 D2).
C1C2 = E, D2 = D2 = E.
We can use the block-diagonal transition matrix T similar (8) and obtain C1 = C2 = E. More-
over, we can assume that D1 and D2 are either diagonal matrices diag (1, —1) or ±E. From the
condition (1) we have
(E 0) C01 D2) = U ?)=9(D1-0) € R
but either the matrix E + 0(D1,0) = 9(D1, E) = 0 or the matrix 6(D1,iE) =0 is singular, it is
impossible. Thus, the conjecture a2 = h1 leads to contradiction, similar for h2.
2. Let a2 = h3. Then a is the matrix (12), up to base changing. The transition matrix
E 0 0 0
T 0 C1 0 0
0 0 E 0
0 0 0 D1
preserves a and maps the Baer involution
/ 0 C1 0 0
C-1 0 0 0
в= 1 0 0 0 D1
V 0 0 D1- 10
to the matrix (12). The condition (1) for 9(V,U) leads to the involutivity m(m(U))
f(f(V)) = V. The theorem is proved.
U,
□
Conclusion
We can see that the properties and the structure of the linear autotopism group for a two-
dimensional semifield plane may be considerably generalized to the N -dimensional case. The
proof technique can be used with more careful consideration.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
In order to study Hughes’ problem on the solvability of the full collineation group of a finite
non-Desarguesian semifield plane, the authors consider it possible to use the obtained results
to further investigations. The method applied will probably be useful to consider simple non-
Abelian groups and to exclude an extensive list from possible autotopism subgroups.
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry
of Science and Higher Education of the Russian Federation (Agreement no. 075-02-2023-936).
References
[1] D.R.Hughes, F.C.Piper, Projective planes, New-York, Springer-Verlag Publ., 1973.
[2] V.D.Mazurov, E.I.Khukhro (eds.), Unsolved Problems in Group Theory. The Kourovka
Notebook, no. 19, Novosibirsk, Sobolev Inst. Math. Publ., 2018.
[3] M.Biliotti, V.Jha, N.L.Johnson, G.Menichetti, A structural theory for two-dimensional
translation planes of order q2 that admit collineation groups of order q2, Geom. Dedicata,
29(1989), 7-43.
[4] O.V.Kravtsova, Semifield planes of even order that admit the Baer involution, The Bulletin
of Irkutsk State University. Series Mathematics, 6(2013), no 2, 26-37 (in Russian).
[5] O.V.Kravtsova, Semifield planes of odd order that admit a subgroup of autotopisms isomor-
phic to A4, Russian Mathematics, 60(2016), no 9, 7-22. DOI: 10.3103/S1066369X16090024
[6] N.D.Podufalov, On spread sets and collineations of projective planes, Contem. Math.,
131(1992), part 1, 697-705.
[7] O.V.Kravtsova, Elementary abelian 2-subgroups in an autotopism group of a semifield pro-
jective plane, The Bulletin of Irkutsk State University. Series Mathematics, 32(2020), 49-63.
DOI: 10.26516/1997-7670.2020.32.49
[8] O.V.Kravtsova, Dihedral group of order 8 in an autotopism group of a semifield projec-
tive plane of odd order, Journal of Siberian Federal University. Mathematics & Physics,
15(2022), no. 3, 21-27.
[9] O.V.Kravtsova, D.S.Skok, The spread set method for the construction of finite quasifields,
Trudy Inst. Mat. i Mekh. UrO RAN, 28(2022), no. 1, 164-181 (in Russian).
DOI: 10.21538/0134-4889-2022-28-1-164-181
[10] H.Luneburg, Translation planes, New-York, Springer-Verlag Publ., 1980.
[11] O.V.Kravtsova, On automorphisms of semifields and semifield planes, Siberian Electronic
Mathematical Reports, 13(2016), 1300-1313. DOI: 10.17377/semi.2016.13.102
[12] N.D.Podufalov, B.K.Durakov, O.V.Kravtsova, E.B.Durakov, On the semifield planes of
order 162, Siberian Mathematical Journal, 37(1996), is. 3, 616-623 (in Russian).
[13] G.E.Moorhouse, PSL(2, q) as a collineation group of projective planes of small orders,
Geom. Dedicata, 31(1989), no. 1, 63-88.
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Olga V. Kravtsova, Daria S. Skok
Linear Autotopism Subgroups of Semifield Projective Planes
[14] O.V.Kravtsova, 2-elements in an autotopism group of a semifield projective plane, The
Bulletin of Irkutsk State University. Series Mathematics, 39(2022), 96-110.
DOI: 10.26516/1997-7670.2022.39.96
[15] M.J.Ganley, Baer involutions in semi-field planes of even order, Geom. Dedicata, 2(1974),
499-508.
[16] N.D.Podufalov, B.K.Durakov, I.V.Busarkina, O.V.Kravtsova, E.B.Durakov, Some results
on finite projective planes, Journal of Mathematical Sciences, 102(2000), no. 3, 4032-4038.
[17] D.R.Hughes, M.J.Kallaher, On the Knuth semi-fields, Internat. J. Math. & Math. Sci.,
3(1980), no. 1, 29-45.
[18] O.V.Kravtsova, Semifield planes admitting the quaternion group Q8, Algebra and Logic,
59(2020), is. 1, 71-81.
Подгруппы линейных автотопизмов
полуполевых проективных плоскостей
Ольга В. Кравцова
Дарья С. Скок
Сибирский федеральный университет
Красноярск, Российская Федерация
Аннотация. Изучается известная гипотеза Д. Хьюза 1959 г. о разрешимости полной группы колли-
неаций недезарговой полуполевой проективной плоскости конечного порядка (также вопрос 11.76
Н. Д. Подуфалова в Коуровской тетради). Эта гипотеза редуцируется к группе автотопизмов, со-
стоящей из коллинеаций, фиксирующих треугольник. В работе описаны элементы порядка 4 и
диэдральные либо кватернионные подгруппы порядка 8 в группе линейных автотопизмов полупо-
левой плоскости ранга 2 над ядром. Основные доказанные результаты являются техническими и
необходимы для дальнейшего изучения подгрупп четного порядка в группе автотопизмов конечной
недезарговой полуполевой плоскости. Результаты могут быть использованы для изучения полупо-
левых плоскостей, допускающих подгруппы автотопизмов из списка Д. Г. Томпсона минимальных
простых групп.
Ключевые слова: полуполевая плоскость, автотопизм, гомология, бэровская инволюция, пробле-
ма Хьюза.
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