UDC 512.41

MATHEMATICS

V. P. SHUNKOV

ON PERIODIC GROUPS WITH CERTAIN FINITENESS CONDITIONS

(Presented by Academician V. M. Glushkov, May 4, 1970)

In accordance with the work \((^1)\), a locally finite group with extremal Sylow \(p\)-subgroups (for a given prime number \(p\)) is called an \(S_pF\)-group. In particular, if a locally finite group is an \(S_pF\)-group for every prime divisor \(p\) of the orders of its elements, then it is called an \(SF\)-group.

In the present paper a result is formulated which gives a necessary and sufficient condition for the invariance of a maximal complete abelian 2-subgroup in an \(S_2F\)-group with an infinite Sylow 2-subgroup (see Theorem 1). Using Theorem 1, it is easy to prove that an arbitrary \(SF\)-group is an extension of an abelian group by means of an \(SF\)-group with finite Sylow 2-subgroups (see Theorem 2). Hence, and from Theorems 3, 4 and a result of S. N. Chernikov \((^2)\), it follows that questions 2.1, 8.1, 8.2 posed in the survey \((^3)\), as well as question 23 from \((^4)\), need only be considered for \(SF\)-groups with finite Sylow 2-subgroups; and since locally soluble \(SF\)-groups have been studied up to \(SF\)-groups with finite Sylow \(p\)-subgroups \((^5)\), and locally finite groups of odd order are locally soluble \((^6)\), the significance of Theorems 2, 3, 4 in connection with the consideration of the indicated questions becomes clear.

With the aid of Theorems 2, 3 the author proves that, for locally finite groups, the minimality condition for abelian subgroups implies the minimality condition for subgroups, and that Sylow \(p\)-subgroups in an \(SF\)-group are conjugate. Consequently, questions 2.1, 8.2 from \((^3)\) are equivalent.

The conjugacy of Sylow \(p\)-subgroups in locally finite groups with the minimality condition for subgroups was first established in the work \((^7)\).

In Section 2 a class of \(p\)-groups is singled out, called by the author biprimitively finite; for \(p = 2\) these coincide with 2-groups, and to this class of groups Blackburn’s result on Chernikov \(p\)-groups \((^8)\) is generalized (see Theorem 8 of the present work).

An example of a group constructed by P. S. Novikov and S. I. Adian \((^9)\) shows that Theorem 8 cannot be generalized to arbitrary \(p\)-groups, even under the condition of finiteness of all abelian subgroups in the group.

At the same time, it follows from a result of E. S. Golod \((^{10})\) that a biprimitively finite \(p\)-group need not be locally finite.

As a corollary of Theorem 8 we obtain that a biprimitively finite \(p\)-group (and for \(p = 2\), an arbitrary 2-group) satisfying the minimality condition for abelian subgroups is a Chernikov \(p\)-group (see Corollary 1).

Corollary 1 for \(p = 2\) was stated without proof by S. P. Strunkov in the abstracts of the Batumi Symposium, 1967, and for \(p\)-groups in which any two elements generate a finite group it was proved by V. G. Viljaper \((^{11})\).

1. A group containing (not containing) involutions will be called a group of even (odd) order.

\(C_G(M)\) is the centralizer of the set \(M\) of elements of the group \(G\).

\(N_G(A)\) is the normalizer of the subgroup \(A\) in the group \(G\).

\(\pi(G)\) is the set of prime divisors of the orders of elements of the group \(G\).

If \(A\) and \(B\) are subsets of the group \(G\), then \(AB\) is the set of elements of \(G\) of the form \(g=ab\) \((a\in A,\ b\in B)\).

\(A\lambda B\) is the semidirect product of the groups \(A\) and \(B\), where \(A\) is a normal divisor in \(A\lambda B\).

An element \(g\) of the group \(G\) is called strictly real with respect to the involution \(i\) from \(G\) if \(igi=g^{-1}\).

By the rank of a group we shall mean the special rank in the sense of Mal’cev \((^{12})\).

Definition 1. A Sylow \(p\)-subgroup \(P\) of a countable locally finite group \(G\) is called projective in \(G\) if the group \(G\) has a chain of finite subgroups

\[ N_1 \subset N_2 \subset \cdots \subset N_s \subset \cdots, \]

whose union coincides with \(G\), such that \(P\cap N_s\) \((s=1,2,\ldots)\) is a Sylow \(p\)-subgroup in \(N_s\).

Every countable locally finite group \(G\), for any \(p\subset \pi(G)\), has a projective Sylow \(p\)-subgroup. This is not hard to prove by using the well-known method of projections \((^{13})\).

Definition 2. We shall say that an \(S_pF\)-group has a \(p\)-complete part if a maximal complete abelian \(p\)-subgroup in it is invariant. In particular, if the Sylow \(p\)-subgroups of an \(S_pF\)-group are finite, then we shall agree to regard the identity subgroup as the \(p\)-complete part.

If \(G\) is an \(S_pF\)-group, then an element \(g\) of order \(p\) from \(G\) will be called an element of the first kind if \(C_G(g)\) has a \(p\)-complete part, and an element of the second kind otherwise.

Definition 3. The rank of the \(p\)-complete part of some projective Sylow \(p\)-subgroup of a countable \(S_pF\)-group \(G\) will be denoted by \(t_p(G)\) and will be called the \(p\)-complete rank of the group \(G\). The index of the \(p\)-complete part in a subgroup \(P\) of the group \(G\) will be called the \(p\)-complete index of the group \(G\) and will be denoted by \(l_p(G)\).

If the Sylow \(p\)-subgroups of the \(S_pF\)-group \(G\) are finite (in particular, if \(G\) contains no elements of order \(p\)), then we shall agree to set \(t_p(G)=0\).

It is easy to prove that the \(p\)-complete rank of a countable \(S_pF\)-group \(G\) does not depend on the choice of the projective Sylow \(p\)-subgroup, and if for some subgroup \(H\) of \(G\) \(t_p(H)=t_p(G)\), then \(l_p(H)\leqslant l_p(G)\), and also the \(p\)-complete rank of a subgroup does not exceed the \(p\)-complete rank of the group.

Theorem 1. Let \(G\) be an \(S_2F\)-group with an infinite Sylow 2-subgroup, satisfying the following condition:

) if \(H\) is a subgroup of odd order and \(N_G(H)\) contains a quasicyclic 2-subgroup \(Q\), then \(Q\subset C_G(H)\). Then a maximal complete abelian 2-subgroup of \(G\) is invariant in \(G\).*

Theorem 1 is proved by contradiction, i.e., it is assumed that there exist \(S_2F\)-groups (without impairing the generality of the argument, they may be assumed countable) satisfying condition *), but not having 2-complete parts.

In the class of all such countable groups one chooses the subset of groups of least 2-complete rank, and in this subset the class of all groups of least 2-complete index; this class is denoted by \(K_2\).

In the first part of the proof of Theorem 1 it is established that in \(K_2\) there exists a group in which every involution is an element of the first kind. Here the main tool of the proof is Glauberman’s theorem \((^{14})\) (and if one does not use this theorem, then Grün’s theorem \((^{15})\)).

In the second part of the proof of Theorem 1, instead of \(K_2\) one considers the subset \(K_2'\) of all groups from \(K_2\) in which every involution is an element of the first kind.

It is proved that \(K_2'\) has a group such that it is either isomorphic to a group of type \(PSL(2,T)\) over a field \(T\) of odd characteristic, or has-

gives a subgroup, some infinite factor group of which is isomorphic to one of the groups: \(SL(2,Q)\), the Suzuki group \(S(Q)\), or the unitary group \(U_3(Q)\), where \(Q\) is a field of even characteristic.

In the first case we obtain a contradiction with condition \(*\), since in a group of type \(PSL(2,T)\) with an infinite Sylow \(2\)-subgroup and over a field of odd characteristic there exists an abelian subgroup \(B\) of odd order, and in its normalizer there is a quasicyclic \(2\)-subgroup which does not centralize the subgroup \(B\).

In the second case we obtain a contradiction with the extremality of the Sylow \(2\)-subgroup in an \(S_2F\)-group, since in infinite groups of type \(SL(2,Q)\), \(S(Q)\), \(U_3(Q)\) over a field of even characteristic, the Sylow \(2\)-subgroups are not extremal.

The main tools in the second part of the proof of Theorem 1 are the result of Gorenstein and Walter \((^{16})\), the result of Suzuki \((^{17})\) on finite groups with an independent Sylow \(2\)-subgroup, Theorem 14.3.1 of \((^{15})\), and the Brauer–Suzuki result \((^{18})\).

Theorem 2. An \(SF\)-group is an extension of an abelian group by an \(SF\)-group with finite Sylow \(2\)-subgroups.

Theorem 3. A locally finite group with the minimality condition for (abelian) subgroups is an extension of an abelian group by a group with finite Sylow \(2\)-subgroups and the minimality condition for (abelian) subgroups.

Theorem 4. If an infinite simple locally finite group of finite rank exists, then its Sylow \(2\)-subgroups are finite.

Theorem 5. For locally finite groups, the minimality condition for abelian subgroups implies the minimality condition for subgroups.

The question of the validity of Theorem 5 for arbitrary periodic groups remains open.

In the proof of Theorem 5, as well as of Theorem 7, together with Theorem 3 the following lemma is used.

Lemma. Let \(G\) be a finite group, \(H\) a proper subgroup of even order coinciding with its normalizer, \(j\in G\) an involution and \(j\notin H\), \(D(j)\) the subgroup of \(H\) generated by all elements which are strictly real with respect to \(j\), and \(i\) some involution from \(H\).

If the subgroup \(H\) is \(2\)-transversal to its conjugates, i.e. intersects any other subgroup conjugate to it in a subgroup of odd order, then

1) the subgroup \(D(j)\) is of odd order, and for every strictly real element \(a\ne 1\) of \(H\) with respect to \(j\), the centralizer \(C_G(a)\) is of odd order;

2) in each adjacent class \(hC_H(i)\) \((h\in H)\) there exists one and only one element from \(D(j)\) which is strictly real with respect to \(j\), and \(H=D(j)C_H(i)\);

3) no Sylow \(p\)-subgroup of \(D(j)\) is centralized by the involution \(j\);

4) if \(D=H\cap jHj\), then \(D\lambda \langle j\rangle\).

Theorem 6. If in a locally finite group the Sylow \(2\)-subgroups are extremal and the subgroups of odd order are finite, then the group itself is extremal.

Theorem 7. The Sylow \(p\)-subgroups of an \(SF\)-group are conjugate.

1. Let \(G\) be a periodic group, and \(p\) some prime number, satisfying the following condition:

If \(H\) is a subgroup of \(G\), \(N\) its invariant extremal subgroup, then in the factor group \(H/N\) any two elements of order \(p\) (if \(H/N\) has such elements) generate a finite group.

In this case we shall call the group \(G\) biprimitively finite relative to \(p\). If \(G\) is biprimitively finite relative to every \(p\in\pi(G)\), then we shall call the group \(G\) biprimitively finite.

Every periodic group of odd order is biprimitively finite with respect to 2.

On the other hand, as the example of E. S. Golod \(^{(10)}\) shows, there exist biprimitively finite \(p\)-groups (for any \(p\)) that are not locally finite groups.

Theorem 8. If a biprimitively finite \(p\)-group (and for \(p=2\) an arbitrary 2-group) possesses a finite maximal elementary abelian subgroup, then the group is extremal.

Corollary 1. If a biprimitively finite \(p\)-group (and for \(p=2\) an arbitrary 2-group) satisfies the minimality condition for abelian subgroups, then the group is extremal.

Corollary 2. A biprimitively finite \(p\)-group (and for \(p=2\) an arbitrary 2-group) is finite if and only if it is generated by a finite number of generators and some maximal elementary abelian subgroup of it is finite.

Institute of Physics
Siberian Branch of the Academy of Sciences of the USSR
Krasnoyarsk

Received
27 IV 1970

CITED LITERATURE

\(^{1}\) V. P. Shunkov, Sibirsk. matem. zhurn., 8, No. 1, 213 (1967).
\(^{2}\) S. N. Chernikov, Matem. sborn., 7, 49, 539 (1940).
\(^{3}\) S. N. Chernikov, UMN, 14, issue 5, 45 (1959).
\(^{4}\) Kourovka Notebook, Novosibirsk, 1965.
\(^{5}\) M. I. Kartapolov, Sibirsk. matem. zhurn., 2, No. 6, 853 (1961).
\(^{6}\) W. Feit, I. G. Thompson, Pacific J. Math., 13, No. 3, 775 (1963).
\(^{7}\) R. Baer, Math. Ann., 150, No. 1, 1 (1963).
\(^{8}\) N. Blackburn, J. Math., 6, 421 (1962).
\(^{9}\) P. S. Novikov, S. I. Adyan, Izv. AN SSSR, ser. matem., 32, No. 1, 2, 3 (1968).
\(^{10}\) E. S. Golod, Izv. AN SSSR, ser. matem., 28, 273 (1964).
\(^{11}\) B. T. Vilyatser, UMN, 13, No. 2, 163 (1958).
\(^{12}\) A. I. Mal'tsev, Matem. sborn., 22, 273 (1948).
\(^{13}\) A. G. Kurosh, Group Theory, 3rd ed., “Nauka,” 1967.
\(^{14}\) G. Glauberman, J. Algebra, 4, 403 (1966).
\(^{15}\) M. Hall, Group Theory, IL, 1962.
\(^{16}\) D. Gorenstein, J. Walter, J. Math., 6, 553 (1962).
\(^{17}\) M. Suzuki, Ann. Math., 80, 58 (1964).
\(^{18}\) R. Brauer, M. Suzuki, Proc. Nat. Acad. Sci. U.S.A., 45, 1757 (1959).