UDC 512.41

MATHEMATICS

V. P. SHUNKOV

ON THE THEORY OF PERIODIC GROUPS

(Presented by Academician A. I. Mal’cev on 9.XI.1966)

1. The question of the existence in groups of infinite proper subgroups is equivalent to the well-known problem of O. Yu. Schmidt on an infinite group with finite proper subgroups. The most substantial results on this question were obtained by S. N. Chernikov \((^1)\) and M. I. Kargapolov \((^2)\).

In the present note it is proved that a periodic group of even order, i.e., a group containing involutions, either coincides with the centralizer of one of its involutions, or it has a proper infinite subgroup. Thus the solution of O. Yu. Schmidt’s problem is reduced to the analogous problem for a periodic group of odd order, i.e., a group containing no involutions.

2. Theorem. An infinite periodic group of even order either coincides with the centralizer of one of its involutions, or possesses a proper infinite subgroup with nontrivial center.

Proof. Let \(G\) be an infinite periodic group of even order and let \(i\) be one of its involutions. If the centralizer \(C_G(i)\) is infinite, then the assertion of the theorem is true.

Let \(C_G(i)\) be a finite group. Denote by \(T\) the set of all nonidentity strictly real elements of odd order of the group \(G\) with respect to the involution \(i\), i.e., such elements \(t \in T\) for which the relation \(i^{-1}ti = t^{-1}\) holds. The set \(T\) is infinite. Indeed, if it were finite, then in the group \(G\) there would exist infinitely many strictly real elements of even order with respect to the involution \(i\). But then, obviously, from the finiteness of the centralizer \(C_G(i)\) it would follow that in the group \(G\) there exists an involution \(i_1\) with infinite centralizer, and in this case the assertion of the theorem is true.

Let \(a\) be some element of \(T\). Then \(a = ik\), where \(k\) is some involution, with \(k \ne i\). From what was proved above it follows that in the set \(T\) there exists only a finite number of elements for which the orders of elements of the form \(u = ib^{-1}kb\), \(b \in T\), are even. Removing such elements from \(T\), we again obtain an infinite set \(M\). Let \(c\) be some element of \(M\). Consider the element \(s = ic^{-1}kc\). Since the order of the element \(s\) is odd, for some element \(s_1\) of \(\{s\}\) we shall have \(s_1^{-1}c^{-1}kcs_1 = i\). On the other hand, for some element \(a_1 \in \{a\}\) we have \(a_1^{-1}ka_1 = i\). From the last two equalities we obtain \(a_1s_1^{-1}c^{-1}kcs_1a_1^{-1} = k\). Consequently, \(cs_1a_1^{-1} = h \in C_G(k)\), \(s_1 = c^{-1}ha_1\). Let \(g = c^{-1}h\). Then \(s_1 = ga_1\). Transform the last equality by means of the involution \(i\): \(i^{-1}s_1i = i^{-1}gii^{-1}a_1i\). In view of the strict reality of the elements \(s_1\) and \(a_1\), we obtain
\(s_1^{-1} = a_1^{-1}g^{-1} = i^{-1}gia_1^{-1}\), or \(a_1i^{-1}gia_1^{-1} = g^{-1}\). Obviously, the element \(j = ia_1^{-1}\) is an involution and \(j^{-1}(h^{-1}c)j = (h^{-1}c)^{-1}\). Since \(c\) is an arbitrary element of \(M\), we have proved that for any element \(c \in M\) there exists an element \(h \in C_G(k)\) such that \(g = hc\) is a strictly real element with respect to the involution \(j = ia_1^{-1}\).

Since the subgroup \(C_G(k)\) is finite, one can extract in \(M\) an infinite sequence of elements \(c_1, c_2, \ldots, c_m, \ldots\) such that for some-

of the element \(r \in C_G(k)\) there will correspond a sequence of strictly real elements \(q_1=rc_1,\ q_2=rc_2,\ldots,\ q_m=rc_m,\ldots\) with respect to the involution \(j=ia_1^{-1}\), \(j^{-1}rc_1j=c_1^{-1}r^{-1}\), \(j^{-1}rc_mj=c_m^{-1}r^{-1}\), \(j^{-1}c_m^{-1}r^{-1}j=rc_m\).

From the first and the last equality we obtain
\(j^{-1}(c_m^{-1}c_1)j=r(c_mc_1^{-1})r^{-1}\), or
\(a_1i^{-1}(c_m^{-1}c_1)ia_1^{-1}=r(c_mc_1^{-1})r^{-1}\) \((j=ia_1^{-1})\).

Since the elements \(c_1\) and \(c_m\) are strictly real with respect to the involution \(i\), from the last equality we obtain
\(r^{-1}a_1(c_mc_1^{-1})a_1^{-1}r=c_mc_1^{-1}\).

Obviously, the set of distinct elements of the form \(c_mc_1^{-1}\) \((m=1,2,\ldots)\) is infinite and, consequently, the centralizer of the element \(d=a_1^{-1}r\) is infinite, while \(C_G(d)\ne G\). Indeed, suppose that \(C_G(d)=G\). Then the involution \(k\) centralizes the element \(d=a_1^{-1}r\). Since \(r\in C_G(k)\), from the permutability of the elements \(k\) and \(d\) we obtain
\(k^{-1}dk=ka_1^{-1}kr=a_1^{-1}r\), or \(k^{-1}a_1^{-1}k=a_1^{-1}\). The last equality is impossible, since, by assumption, \(a_1^{-1}ka_1=i\ne k\). The theorem is proved.

Corollary 1. There is no infinite simple group of even order all of whose proper subgroups are finite.

Corollary 2. It suffices to solve O. Yu. Schmidt’s problem for periodic groups of odd order.

Corollary 3. The question of the existence of an infinite abelian subgroup in arbitrary periodic groups reduces to the analogous question for periodic groups of odd order.

Since a group of odd order is soluble \((^3)\), in view of the results of S. N. Chernikov \((^1)\), the theorem proved implies the previously known result \((^2)\):

An infinite locally finite group has a proper infinite abelian subgroup.

Krasnoyarsk Branch
of Novosibirsk State University

Received
24 X 1966

REFERENCES

\(^1\) S. N. Chernikov, Matem. sborn., 7, 49, 539 (1940).
\(^2\) M. I. Kargapolov, Sibirsk. matem. zhurn., 4, No. 1, 232 (1963).
\(^3\) W. Feit, J. G. Thompson, Proc. Nat. Acad. Sci. U. S. A., 48, 6, 968 (1962).