On sharply doubly-transitive groups Текст научной статьи по специальности «Математика»

УДК 512.145

On Sharply Doubly-Transitive Groups

Evgeny B. Durakov* Evgenia V. Bugaeva^ Irina V. Sheveleva*

Institute of Core Undergraduate Programmes, Siberian Federal University, Kirensky, 26, Krasnoyarsk, 660074,

Russia

Received 08.07.2012, received in revised form 20.09.2012, accepted 20.12.2012 The question of existence in any sharply doubly-transitive group of a regular abelian normal subgroup is investigated.

Keywords: group, sharply doubly-transitive group, near-field near-domain.

The permutation group G acting on a set Q (|Q| > 3) is called sharply doubly-transitive if for any two ordered pairs of elements from the set Q there exists a unique element from G, mapping the first pair to the second one. In particular, the only permutation in G stabilizing two points is the identity.

C.Jordan [1] (according to [2]) proved that in a finite sharply doubly-transitive group all the regular permutations together with the identity one form an abelian normal subgroup. It is still not known if it is true in the infinite case. Sharply doubly-transitive groups are closely connected with near-fields and near-domains. Near-fields were introduced by L.E.Dickson in 1905 [3] as algebraic systems (F, +, •) with two binary operations, where

1. (F, +) is an abelian group with the identity element 0,

2. (F*, •) is a group with the identity element 1 (F* = F \ {0}),

3. Multiplication distributes over addition on the left, viz. a • (b + c) = a • b + a • c.

R.D.Carmichael [4] established that the groups T2(F) of affine transformations x — a + bx (b = 0) of a near-field F are sharply doubly-transitive. H.Zassenhauz [5] classified finite sharply

doubly-transitive groups and the corresponding near-fields. M.Holl [6] revealed a relation between

these groups and projective planes, and under additional restrictions he generalized the Jornan’s theorem on the infinite case (theorem 20.7.1 [7]).

J.Tits [8], Gratzer [9] and H.Karzel [10] introduced a more general, than the near-fields, algebraic systems (F, +, •) later called near-domains, in which (F, +) is a loop, (F*, •) is a group, and for any elements a, b, c G F the following conditions hold:

1. a + b = 0 —— b + a = 0;

2. 0 • a = 0 — a • 0 = 0;

* durakov@mail.ru

tbugaevaevgeniya@mail.ru

tshiv@krasmail.ru © Siberian Federal University. All rights reserved

3. а • (b + c) = а • b + а • c;

4. There exists a unique element da,b Є F* such that а + (b + x) = (а + b) + da,b • x for each x Є F.

Thus, for every sharply doubly-transitive group T there exists a corresponding near-domain F, for which T is the group of affine transformations x ^ а + bx (b = 0) [2], and conversly, the group T2(F) of the affine transformations of a near-domain F is sharply doubly-transitive on F. The existence of a near-domain which is not a near-field is the central problem of the theory of these algebraic systems. In particular, the foresaid problem of the existance of a regular abelian normal subgroup of a sharply doubly-transitive group is equivalent to the problem of coincidence between near-domains and near-fields.

V.D.Mazurov [11] proved that in every sharply doubly-transitive group G of the zero characteristic the stabilizer of any point contains a subgroup, which is isomorphic to the multiplicative group of the field of rational numbers, and in the group G there is a subgroup, which is isomorphic to the affine group of the stabilizer. For sharply doubly-transitive groups of an odd characteristic p the following theorem is proved:

Theorem І. Let T be a group of affine transformations of a near-domain F of an odd characteristic p, j be an involution from the stabilizer Ta of a point а Є F, b be an element from T \ Ta, strongly real with respect to j; let A = CT(b) and V = NT(A). Then

Ї. The subgroup A is inverted by j and it is also strongly isolated in T;

2. The subgroup V acts sharply doubly-transitive on the orbit Д = аА; A is an Abelian regular normal subgroup in V, H = V П Ta is the stabilizer of a point and V = A X H.

In the 14th edition of the Kourovka Notebook V.D.Mazurov stated a question 12.48(a) [12]

about the existence of a regular Abelian normal subgroup in a sharply doubly-transitive group

with a locally finite stabilizer of a point. In a special case the following theorem gives an affirmative answer to this question:

Theorem 2. Let T be a sharply doubly-transitive subgroup, let a near-domain F be its locally-finite subgroup, containing a regular permutation and let its intersection with a stabilizer of some point be normal in this stabilizer and contain more than two elements. Then the group T has a regular normal abelian subgroup, and a near-domain F(+, •) is a near-field of a non-zero characteristic.

Results were announced in [13].

Proof of the theorem Ї. Due to the conditions of the theorem, the group T is sharply doubly-transitive on F (ex. V.1.2 [2]) and the stabilizer Ta of a point а Є F contains an involution j (ex. V.1.4 [2], a lemma 3.4 [14]). According to ex. V.1.3 [2] and a lemma 3.4 [14], the involution j is unique in Ta, in particular Ta = CT(j). It means that the group T acts (by conjugation) sharply doubly-transitively on the set J of its involutions, and Ta acts transitively on the set J \ {j} (a lemma 3.4 from [14]). Hence all the elements vk (v, k Є J, v = k) are conjugate in T, in particular, all non-identity elements vk Є J2 have the same order which is either infinite or equal to a prime number p = 2. In the former case the characteristic of the near-domain F and the group T is equal to 0, in the latter case we have Char F = Char T = p. Due to the conditions of the theorem only the latter case Char F = Char T = p takes place.

Let Nj = jJ and Nj* = Nj \ {1}. Since Ta acts transitively on the set J \ {j}, all the

elements of the form jk (k G J, k = j) are conjugate by suitable elements from Ta and Ta acts transitively on the set N*. Due to the properties of the dihedral groups (lemma 2.10 [14]) the set N* coincides with the set of all the elements from T \ Ta strongly real with respect to j. Besides, if c G N* then any non-identity element of a cyclic subgroup (c) is contained in N*.

Let b G T \ Ta and bj = b-1. Then from the above reasoning it follows that b G Nj and j /A = CT(b). Clearly j G V = NT(A), and for (T, Ta) is Frobenius pair (lemma 3.4 [l4]) we have A n Ta = 1 and CA(j) = 1. Recall that the involution j of the group T is called finite if for any t G T the subgroup (j, j4) is finite, that is equivalent to the finiteness of orders of the elements jj4. Due to the foresaid, the involution j is finite in the group T and hence it is finite in the subgroup K = A X (j). By lemma 2.20 [14] the subgroup A is Abelian, it is inverted by the involution j, and it is clear that A C Nj. Then, each non-identity permutation from T is either regular or stabilizes only one point from F. It follows that the permutation b is regular and each non-identity element from A is as well regular on F.

Now we prove that the subgroup of A is strongly isolated in T, that is, it contains a centralizer of any not identity element. Let c be any non-identity element from A and C = CT(c). It is clear that A < C and, as it is proved above, c G T \ Ta and cj = c-1. It means that the statements true of the subgroup A are also true of C. In particular, the subgroup C is Abelian. Hence, C < CT(b) = A, and for c G A# is arbitrary it follows that the subgroup of A is strongly isolated in T.

Thus, A C Nj. Since the Ta action by conjugation is transitive on a set Nj*, for any nonidentity elements b, c G A there exists an element h G Ta such that bh = c. Therefore, c G A n Ah and for A is strongly isolated in T and A is commutative we have A, Ah < CT(c) = A and A = Ah. It follows that h G Ta n NT(A) = H and the action of H on A# is transitive.

Next we show that A = aA is the orbit of the group V = A X H. Indeed, if [ G A, then

[ = ab for a suitable element b G A, and for any h G H we have

[h = abh = ah~lbh = ac,

where c = bh G A. Thus we have [h G A for any [ G A, h G H, and A = aV. Further, if [, y G A, and [ = a = 7, [ = 7, then [ = ab, [ = ac, for suitable elements b, c G A#. As it is shown above, bh = c for some h G H. But then [h = abh = ah bh = ac = 7 and H acts transitively on A \ {a}. Thus, V is doubly-transitive on A and for T acts sharply doubly-transitivity on F, the subgroup V acts sharply doubly-transitive on A. Hence, the subgroup A is Abelian, regular on A and it is normal in the group V. □

Proof of the theorem 2. Let L be a locally finite subgroup of the group T, b be a regular permutation from L and Ta be the stabilizer of the point a G F for which a subgroup of H = L n Ta is normal in Ta and |H | > 2. It is clear that H is the proper subgroup in L and due to lemma 3.4 [14] (L, H) is a Frobenius pair. Since L is locally finite, L = M X H is a Frobenius group with a kernel M and a complement H, thus the set of the regular permutations from L coincides with M#. Let t G M#. Then t / Ta and due to the above reasoning, the subgroup of St = (H, H4) is a Frobenius group with a kernel M n St and a complement H. By conditions of the theorem, H is a normal subgroup Ta and for each element h G Ta the subgroup Sth = Sh = (H, Hth) is a locally finite Frobenius group with a complement H.

Since |H| > 2 and Ta can not contain more than one involution (a lemma 3.4 [14]), there

exists an element a in H, whose order is > 2. Now we prove that for any element g G T \ Ta the subgroup Lg = (a, ag) is a Frobenius finite group with a complement Hn Lg. Again, let t G M#.

As a doubly-transitive group, T is the union of the subgroup T^ and the double cosets TatTa. In particular, the element g is representable in the form g = rth, where r, h Є Ta. Since аг Є H, Lrt = (а, а14) is a subgroup of the group St. It follows that Lg < Sth and Lg is a Frobenius finite group with a complement H П Lg.

Thus, the group T, its proper subgroup Ta and the element а Є Ta satisfy all the conditions

of theorem 2.11 from [15]. That theorem gives T = A X Ta.

It is left to make sure that A is a regular Abelian subgroup. Suppose Ta to contain an involution j. As all the involutions in T are conjugate, and j is the only involution in Ta, the set J of all the involutions of the group T is contained in the coset Aj. According [14, lemma 3.4, item 3] the involution j is perfect in G, and as J С A X (j), j is perfect in A X (j). Due to lemma 2.19 [14] A is an Abelian group inverted by the involution j, besides, Ta acts transitively on A# by conjugation. Due to an obvious inclusion M < A, A is an elementary Abelian p-group

and Char T = p. The theorem for this case is completely proved.

Now suppose that Ta doesn’t contain an involution. By lemma 3.3 [14] the set J of all involutions of the group T is not empty and it is clear that J С A. Since T is sharply doubly-transitive on F, for any point в Є F there exists an involution k such that ак = в. Therefore, TatTa = Ta J, J = A# and A is a group of period 2. Hence A is an Abelian group and Char T=2. □

The authors are grateful to professor A.I.Sozutov for the statement of the problem and constant attention to the work.

The work is supported by the RFBR (grant no. ЇО-ОЇ-00609-a).

References

[1] C.Jordan, Sur la classificfniondes groupes primitifs, G. R. Acad. Sc., 73(1871), 853-857.

[2] H.Wahling, Theorie der Fastkorper, Essen, Thalen Ferlag, 1987.

[3] L.E.Dickson, On finite algebras, Nachr. Acad. Wiss. Gottingen, Math-Phys Kl. II, (1905), 358-393.

[4] R. D.Carmichael, Algebras of certain doubly transitive groups, Amer. J. Math., 53(1931), 631-644.

[5] H.Zassenhaus, Kennzeichnung endlicher linearen Gruppen als Permutationsgruppen, Abh. Math. Sem. Univer. Hamburg ЇЇ, (1936), 17-40 (Dissertation 1934).

[6] M.Hall, Proective planes, Trans. Amer. Math. Soc., 54(1943), 229-277.

[7] M.Hall, Theory of groups, Moscow, 1962 (in Russian)

[8] J.Tits, Generalization des groupes proectifs, Acad. Roy. Belg. Gl. Sci. Mem. Goil. б Ser.,

35(1949), 197-208, 224-233, 568-589, 756-773.

[9] G.Grotzer, A theorem on doubly transitive permutation groups with applicftion to universal algebras, Fund. Math., 53(1963), 25-41.

[10] H.Karzel, Inzidenzgruppen. Vorlezungausarbeiten von I.Peiper und K. Sorensen, Univ. Hamburg, 1965.

[11] V.D.Mazurov, On sharply doubly-transitive groups, Algebra and logic questions, Novosibirsk, 1996, 233-236 (in Russian).

[12] The Kourovka Notebook: Unsolved problems in group theory. 6-17-th Ed., Novosibirsk, 1978-2012 (in Russian).

[13] A.I.Sozutov, E.V.Bugaeva, I.V.Busarkina, On sharply doubly-transitive groups, Algebra, logic and appendices, Krasnoyarsk, 2010, 83-85 (in Russian).

[14] A.I.Sozutov, N.M.Suchkov, N.G.Suchkova, Infinite groups with involyution, Krasnoyarsk, SFU, 2011 (in Russian).

[15] A.M.Popov, A.I.Sozutov, V.P.Shunkov, Groups with systems of Frobenius subgroups, Krasnoyarsk, 2004 (in Russian).

О точно дважды транзитивных группах

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