UDC 519.46

MATHEMATICS

Yu. N. Mukhin

ENDOMORPHISMS OF THE LATTICE OF CLOSED SUBGROUPS OF A TOPOLOGICAL GROUP

(Presented by Academician V. M. Glushkov on 20 III 1969)

The set \(L(G)\) of all closed subgroups of a topological group \(G\) is a complete lattice (structure) with respect to intersection and the taking of the smallest closed subgroup \(A \vee B\) containing \(A, B \in L(G)\). For the case of a discrete group \(G\), an extensive literature is devoted to the relations between the structure of \(G\) and \(L(G)\) (see, for example, the monograph of M. Suzuki \((^1)\)). Homomorphisms of \(L(G)\) have been studied deeply, in particular \(( (^1),\) Chapter III). The present note is devoted to analogous problems for locally compact topological groups. Namely, the question is studied under what conditions the mapping \(X \to X \vee S\), where \(S\) is a proper closed subgroup of \(G\), is an endomorphism of the lattice \(L(G)\). A complete solution of this problem in the discrete case was given by G. Pappa \((^2)\), D. Higman \((^3)\), and S. Sato \((^4)\). Since

\[ (X \cap Y) \vee S = (X \vee S) \cap (Y \vee S), \]

if \(X \to X \vee S\) is an endomorphism, the question turns out to be closely connected with distributive relations between \(\cap\) and \(\vee\). The distributive law in \(L(G)\) was studied by us in \((^5)\), where analogues were obtained of the classical results of O. Ore \((^6)\) on groups with a distributive lattice and on distributive pairs of subgroups for topological groups. All groups considered below are assumed to be locally compact, which is not stipulated in the formulations.

A pair \(A, B\) of elements of \(L(G)\) is called distributive (dually distributive) if, for every \(X \in L(G)\), the distributive law holds

\[ (A \vee B) \cap X = (A \cap X) \vee (B \cap X) \]

(respectively, the dual equality). These notions, introduced by O. Ore, were studied in \((^{6,7,5})\). We have obtained \((^5)\) a criterion for distributivity of a pair of closed subgroups, playing an important role in the proof of the following theorems. We give it here in the following form.

Theorem 1. For a pair \(A, B\) of closed subgroups of a group \(G\) to be distributive, it is necessary that:

(1) the connected compact subgroups of \(A \vee B\) lie in \(A \cap B\);

(2) all primary elements of \(A \vee B\) lie in \(A \cup B\);

(3) every pure element of \((A \vee B) \setminus (A \cup B)\) have finite and relatively prime orders with respect to \(A\) and \(B\),

and it is sufficient that conditions (2) and (3) be fulfilled.

Recall that an element \(x\) is called primary with respect to a prime number \(p\), or a \(p\)-element, if \(\lim x^{p^n} = 1\) as \(n \to \infty\). By \(\Pi(M)\) is denoted the set of all prime numbers \(p\) for which the subset \(M\) of \(G\) contains a \(p\)-element \(\ne 1\). An element \(x\) is pure if the cyclic subgroup \(\{x\}\) generated by it is infinite and closed. A group \(G\) is periodic if it has no pure elements.

An element \(S \in L(G)\) forming a distributive pair with every element of \(L(G)\) is called standard. The notion of a standard element-

introduced in (8). It is also established there that if \(S\) is standard, then \(X \to X \vee S\) is a homomorphism, and dually. It is easy to verify that if \(N \in L(G)\) is simultaneously standard and dually standard, then \(N\) generates a distributive sublattice, i.e., is a neutral element in \(L(G)\). The neutral elements in the lattice of a finite group were described by M. Suzuki (9) and G. Zappa (10), thereby solving problem 35 of G. Birkhoff (11). Below we shall obtain a far-reaching generalization of this result. A neutral element of \(L(G)\) possessing a complement is called central. The following theorem describes the central elements, serves as a criterion for the reducibility of \(L(G)\), and at the same time gives a criterion for the distributivity and dual distributivity of a pair of mutually prime subgroups. It contains Jones’s theorem ((1), p. 19), as well as Theorem 2 from (7).

Theorem 2. Let \(A\) and \(B\) be proper closed subgroups of a group \(G\) such that \(A \vee B = G\), \(A \cap B = E\). Then the following conditions are equivalent:

(1) the pair \(A,B\) is distributive in \(L(G)\);
(2) the pair \(A,B\) is dually distributive in \(L(G)\);
(3) \(L(G) = L(A) \times L(B)\);
(4) the group \(G\) is zero-dimensional, periodic, and decomposes into the topological direct product \(G = A \times B\), with \(\Pi(A) \cap \Pi(B) = \varnothing\).

Suppose now that \(X \to X \vee S\) is a proper endomorphism of the lattice \(L(G)\), i.e., \(S \ne G\) and \(S \ne E\). Then the following five propositions hold, with the help of which our results are obtained. As in the discrete case ((1), p. 100), it is proved that:

\(1^\circ\). \(S\) is a normal subgroup of \(G\).

\(2^\circ\). If \(H \in L(G)\) and the subgroup \(HS\) is closed, then the pair \(H,S\) is distributive.

It follows from this that \(S\) is standard if \(S\) is compact or open. In particular, for discrete groups the concepts of a standard subgroup and of the upper kernel of the endomorphism \(X \to X \vee S\) coincide (S. G. Ivanov). In the general case, the coincidence of these two concepts can be established only after a description of the endomorphisms of \(L(G)\).

\(3^\circ\). The factor group \(G/S\) is periodic.

\(4^\circ\). \(S\) is incident with the connected component \(G_0\) of the identity in \(G\).

\(5^\circ\). If in \(G \setminus S\) there is a \(p\)-element \(x\), then \(S\) is zero-dimensional, elementwise permutable with \(x\), all \(p\)-elements of \(S\) lie in \(\{\bar{x}\}\), and the group \(G\) is periodic.

We give a partial converse to proposition \(5^\circ\), which nevertheless contains Theorem III. 6 from (1).

Theorem 3. A compact or open proper subgroup \(S\) of a zero-dimensional periodic group \(G\) is standard in \(L(G)\) if and only if, for every \(p\)-element \(x\) of \(G \setminus S\), all \(p\)-elements of \(S\) lie in \(\{\bar{x}\}\), and \(x\) belongs to the centralizer of the subgroup \(S\) in \(G\).

By somewhat narrowing the class of periodic groups, one can obtain a sharper result. We shall call a group \(G\) inductively compact if every finite set of its elements lies in a compact subgroup (i.e., \(G\) is equal to the inductive limit of compact groups). In a zero-dimensional inductively compact group all topological Sylow \(\Pi\)-subgroups, for any set \(\Pi\) of prime numbers, turn out to be closed. By \(Q_p\) (\(J_p\)) we denote the additive group of all (all integral) \(p\)-adic numbers, and by \(D_p\) the discrete group of type \(p^\infty\). Let the group \(Q\) be generated by the elements \(b, a_1, a_2, \ldots, a_n, \ldots\), with relations \(a_n^2 = a_{n-1}\), \(a_1^2 = 1\), \(b^2 = a_1\), \(b^{-1}a_n b = a_n^{-1}\); the non-Abelian subgroups of \(Q\) are called generalized quaternion groups.

Theorem 4. Let \(G\) be a zero-dimensional inductively compact group. If \(X \to X \vee S\) is a proper endomorphism of \(L(G)\), then \(G = H \times K\), \(\Pi(H) \cap \Pi(K) = \varnothing\), \(S = H \times A\), where \(A\) is a central subgroup of \(G\) lying in \(K\), and for every \(p \in \Pi(A)\) every Sylow \(p\)-subgroup of \(K\) is either cyclic

or one of the types \(D_p, J_p, Q_p\), or the discrete generalized quaternion group. Conversely, in a group of the indicated structure, \(S\) is a standard subgroup.

No less strong a restriction is the presence of a standard subgroup in an aperiodic or nonzero-dimensional group.

Theorem 5. Let \(G\) be an aperiodic zero-dimensional group. If \(X \to X \vee S\) is a proper endomorphism of \(L(G)\), then:

(1) \(S\) is an open normal subgroup in \(G\);

(2) \(G/S\) is periodic and \(\Pi(S) \cap \Pi(G/S)=\varnothing\);

(3) if \(x \in G \setminus S\), then \(x\) is pure, and for every \(s \in S\) there exists an integer \(n\) such that \(x^n \in xS,\ x^n s=sx^n\);

(4) in \(G\) there are no mutually coprime discrete infinite cyclic subgroups.

Conversely, if \(G\) satisfies conditions (1)—(4), then \(S\) is standard in \(L(G)\).

This theorem contains Theorem III.7 from \((^1)\) and is proved analogously. The following theorem has no analogues for discrete groups.

Theorem 6. Let the group \(G\) be nonzero-dimensional. If \(X \to X \vee S\) is a proper endomorphism of \(L(G)\), then \(G\) is periodic and belongs to one of the following types:

I. \(G_0\) is isomorphic to the special unitary group \(SU(2)\) of complex matrices of order two; \(S=\{s\}\) coincides with the center of \(G_0\), \(s^2=1\); every 2-element of \(G\) has finite order modulo \(S\).

II. \(G_0\) is a one-dimensional compact group (“solenoid”); \(S=\{s\}\subseteq G_0\), \(s^2=1\), \(S\) is central in \(G\); all Sylow 2-subgroups of \(G\) are infinite quaternion groups.

III. \(G_0\) is a solenoid; \(S\) is a zero-dimensional subgroup of \(G_0\), central in \(G\); \(\Pi(S)\cap \Pi(G/G_0)=\varnothing\).

Conversely, in groups of the indicated types, \(S\) is a standard subgroup.

With the aid of Theorems 4 and 6, for a sufficiently broad class of locally compact groups an analogue of Problem 35 from \((^{11})\) is solved.

Theorem 7. Let the group \(G\) be compact mod \(G_0\). \(L(G)\) has a proper neutral element \(N\) if and only if \(G\) is compact and one of the following conditions is fulfilled:

(1) \(G_0\) is a solenoid, \(G=\{b\}\cdot C,\ b^4=1\), where \(C\) is the centralizer of \(G_0\) in \(G\), \(N=\{b^2\}\subseteq G_0\);

(2) \(G_0\) is a solenoid and lies in the center of \(G\), \(N\) is a zero-dimensional subgroup in \(G_0\), \(\Pi(N)\cap \Pi(G/G_0)=\varnothing\);

(3) \(G\) is zero-dimensional and is equal to \(H\times (B\ltimes M)\); \(\Pi(H)\), \(\Pi(B)\), and \(\Pi(M)\) are pairwise disjoint; \(N=H\times A\); \(A\) is central in \(G\) and \(\Pi(A)=\Pi(B)\); \(B\) is a metabelian group, each primary component of which is either cyclic, or \(J_p\), or a generalized quaternion group.

From item (3) of Theorem 7 follows the result of M. Suzuki and T. Tsappa \((^1,\) Theorem III.12), on which our proof does not rely. In the proofs we used certain results of V. P. Platonov \((^{12})\) and D. Lee \((^{13})\).

Ural State University
named after A. M. Gorky
Sverdlovsk

Received
20 III 1969

REFERENCES

\(^1\) M. Suzuki, The structure of a group and the structure of its lattice of subgroups, IL, 1960.
\(^2\) G. Zappa, Giorn. Mat. Battaglini, 78, 182 (1949).
\(^3\) D. G. Higman, Proc. Am. Math. Soc., 2, 467 (1951).
\(^4\) S. Sato, Osaka Math. J., 4, 229 (1952).
\(^5\) O. H. Mukhin, Sibirsk. Mat. Zh., 8, 366 (1967).
\(^6\) O. Ore, Duke Math. J., 4, 247 (1938).
\(^7\) P. G. Kontorovich, S. G. Ivanov, G. P. Kondrashov, DAN, 160, 1001 (1965).
\(^8\) G. Grätzer, E. Schmidt, Acta Math. Acad. Sci. Hung., 12, 17 (1961).
\(^9\) M. Suzuki, Trans. Am. Math. Soc., 70, 372 (1951).
\(^ {10}\) G. Zappa, Rend. Acad. Sci. Fis. Mat. Napoli, 18, 22 (1951).
\(^ {11}\) G. Birkhoff, Lattice Theory, IL, 1952.
\(^ {12}\) V. P. Platonov, Sibirsk. Mat. Zh., 7, 854 (1966).
\(^ {13}\) D. H. Lee, Math. Zs., 104, 28 (1968).