Maximal Compact Normal Subgroups and Pro-lie Groups

We are concerned with conditions under which a locally compact group G has a maximal compact normal subgroup K and whether or not G/K is a Lie group. If G has small compact normal subgroups K such that G/K is a Lie group, then G is pro-Lie. If in G there is a collection of closed normal subgroups {Ha} such that f~| Ha = e and G/Ha is a Lie group for each a, then G is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group. DEFINITION 1. A locally compact topological group G is an L-group if, for each neighborhood U of the identity and compact subset C of G, there is a neighborhood V such that gHg-1 n C C U for every g G G and subgroup H c V. DEFINITION 2. A topological group G is pro-Lie if G has small compact normal subgroups K such that G/K is a Lie group. DEFINITION 3. A locally compact group G is a residual Lie group if there is a collection of closed normal subgroups {Ha} such that f]Ha = e and G/Ha is a Lie group for each a. It is easy to see that a residual Lie group is an L-group [1, Theorem 1.3 and its Corollary]. In this paper we obtain the following: If G is an L-group and has an open normal subgroup Gi such that Gi/Gq is compact, then G is pro-Lie. There exists a locally compact group G with a maximal compact normal subgroup K such that G/K is not a Lie group. A residual Lie group is not necessarily embeddable by a continuous isomorphism in a pro-Lie group. A generalized FC-group has a maximal compact normal subgroup. PROPOSITION 1. For a locally compact group G, the following are equivalent: (1) G has a compact normal subgroup K such that G/K is a Lie group. (2) G has a compact normal subgroup K such that G/K is locally connected. (3) G has an open normal subgroup Gi such that Gi/Gq is compact. We denote the connected component of the identity of G by GoPROOF. Obviously (1) implies (2). We assume that K is compact normal and G/K is locally connected. Thus iG/K)0 = Go/K is open in G/K. It follows that Ä"Go = Gi is an open normal subgroup of G and Gi/Gq is compact. We complete the proof by showing that (3) implies (1). Assuming (3) we have a maximal compact normal subgroup K of Gi and Gi/K is a Lie group [1, Proposition 2.6]. Since K is maximal in Gi and Gi is normal in G, K is normal in G. Since G/Gi is discrete and Gi/K is a Lie group, G/K is a Lie group. Received by the editors January 11, 1984. 1980 Mathematics Subject Classification. Primary 22D05. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 373 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 374 R. W. BAGLEY AND T. S. WU THEOREM 2. For a locally compact group G, the following are equivalent: (1) G is pro-Lie. (2) G is an L-group and G/Gq is pro-Lie. (3) G is an L-group and has a compact normal subgroup K such that G/K is locally connected. (4) G is an L-group and has an open normal subgroup Gi such that Gi/Gq is compact. PROOF. Obviously (1) implies (3). The reverse implication follows from Proposition 1 and [1, Theorem 1.2 and Corollary 2]. Also by Proposition 1, (3) and (4) are equivalent. Since (1) obviously implies (2) we can complete the proof by showing that (2) implies (4). This follows immediately from [1, Proposition 1.7 and Corollary 2]. If G is a locally compact group and K is a maximal compact normal subgroup, then there are some obvious cases when G/K is a Lie group. For example: (1.) If there is an open normal subgroup Gi such that Gi/Gq is compact, then G/K is a Lie group. Now Gi has a maximal compact normal subgroup Ki such that Gi/Ki is a Lie group. Also Ki is normal in G; thus, G/Ki is a Lie group. It follows that, if K is maximal compact normal in G, then G/K is a Lie group. (2) If G/H is locally connected for some compact normal subgroup H, then G/K is a Lie group. (3) If G/K is pro-Lie, then it is a Lie group. By the semidirect product H xn K of two groups H and K we mean the group determined by a homomorphism r\:K —> AiH), the automorphism group of H, with the group operation in H xn K defined by (fo,fc)(/ii,fci) = (/ii/(A;)(/ii),kki). EXAMPLE 1. Let G = iY^7TM-i Ki) x ([]*! Ki) xv Z, where K% = {0,1} and the integer n G Z shifts coordinates n places to the left. We assume ¿^Ki nas the discrete topology and \\K{ has the product topology. It is easy to see that any nontrivial invariant subset C of G contains elements (fc(n),¿rJ), where the nth coordinate of fc(n) is 1 for infinitely many integers n, and j is some fixed integer. Thus C is not compact. It follows that the maximal compact normal subgroup of G is the identity. Thus we have a compactly generated group with maximal compact normal subgroup K such that G/K is not a Lie group. There are many examples of topological groups without maximal compact normal subgroups; for example, any discrete group with no maximal finite subgroup. EXAMPLE 2. We give another example of a locally compact group G which is not a Lie group and in which the identity is the maximal compact normal subgroup. Let G = Ylili Hi xn rii^o^' where Bx = Z, the additive group of integers, and Ki = Z2, the multiplicative two-element group. The automorphism 77(fc) of ^ -Ö» is defined as r/(fc)(/i)j = fc¿/i¿. It is easy to see that if J^Hi has the discrete topology and H Ki the product topology, G has the desired properties. DEFINITION. A locally compact group G is a generalized FC-group if G = Ai D Ai D ■ ■ ■ D An D An+i = e, where each A¿ is a closed normal subgroup of G and Ai/Ai+i is a compactly generated FC-group. We note that by [2, Corollary 3.9], every compactly generated FC-group has a compact normal subgroup such that the corresponding factor group is of the form V xZk. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use MAXIMAL COMPACT NORMAL SUBGROUPS 375 THEOREM 3. If G is a locally compact generalized FC-group, then G has a maximal compact normal subgroup. PROOF. We proceed by induction, noting that, for n = 1, the conclusion follows from the corollary referred to above. Let G — Ai D A2 D • • • D An+i = e, where each Ai/Ai+i is a compactly generated FC-group. We assume that every generalized FC-group with sequence {Ai} of shorter length has a maximal compact normal subgroup. Thus G/An has a maximal compact normal subgroup M and An has a maximal compact normal subgroup K. Let N = 7r_1(M), where it is the canonical mapping of G onto G/An. Again, using [2, Corollary 3.9], we let Q = ZiAn/K,N/K) and note that Q is a compactly generated central group. Thus Q has a maximal compact normal subgroup. We let P be the inverse image of this subgroup under the canonical mapping of N onto N/K. We show that P is the maximal compact normal subgroup of G. Let L be any compact normal subgroup of G. Then 7r(L) C M and L c N. Also L/K n An/K = {e} since An/K SS V x Zk. If follows that L/K C Q and L C P as desired. DEFINITION 4. A topological group G is an N-group if for every neighborhood U of the identity and compact set G of G, there is a neighborhood V of the identity such that gVg-1 C t/U(G-G) for all g G G. A group G is a SIN-group if G has small neighborhoods of the identity which are invariant under the inner automorphisms ofG. It is easy to see that, if a topological group G can be embedded in an SIN-group, then G is an ./V-group. EXAMPLE 3. We construct an example of an iV-group which is residual Lie and which cannot be embedded in a pro-Lie group. We begin with the group F of 2 X 2 matrices of determinant equal to 1. Let H be the subgroup of F of matrices of the form ( ¿ j ) • If A^o is a normal subgroup of F x F which contains all elements of the form [A, A'1], then iV0 contains all pairs of the form [I, C], where / is the identity and G is an element of H. To show this, let A = (%v y)a) and B = (^ °). Then i/ya\ (y-i/a [AB,A^B0 y)'\0 1/y is in Nq. Thus, if , a2 — 1/ca a \ , _ fca/a2 — 1 -1/a A— { ' . „ . and B 0 ca/a2-lj V 0 a2 -l/ca, then [A,B] is in Nq. It follows that [/, (¿ cx)] is in -/V0 as desired, since [/, (¿ [)] — [A,B\[A-\A\. If c > 0 and C = (¿ J), then [J,G] is the conjugate of [/, (¿ })] by [A"1, A], where