On Torsion Subgroups of Lie Groups

We are concerned with torsion subgroups of Lie groups. We extend the classical result of C. Jordan on the structure of finite linear groups to torsion subgroups of connected Lie groups. 1. In [1], Boothby and Wang proved that, for any connected Lie group G, there exists a number k{G) such that any finite subgroup contains an abelian normal subgroup whose index is bounded by k{G), thereby generalizing the famous result of C. Jordan for GL{n,C). Moreover, by using an integral formula of Weyl, they described the bound explicitly. The main purpose of this note is to extend their result to torsion subgroups of connected Lie groups by using the same bound k{G) as above. In order to describe our result, we first recall the bound of index k{G) as presented in [1]. Let G be a connected Lie group and K a maximal compact subgroup of G. Let % denote the Lie algebras of K and let Q be the set consisting of all X E % such that the absolute values of the eigenvalues of ad X are all less than tt/6, and let U = expK Q. Then k{G) is defined to be p{K)/p{U) where p is an invariant Haar measure of K. That k{G) does not depend on the choice of u is clear. Then our main result states: Theorem. Let G be a connected Lie group. Then every torsion subgroup of G contains an abelian normal subgroup whose index is bounded by k{G). The following lemmas are needed for the proof of the theorem. Lemma 1 (Selbert [2, p. 154]). Every finitely generated subgroup ofGL{n, C) contains a normal torsion free subgroup of finite index. Lemma 2. Every torsion subgroup of a connected Lie group is contained in a maximal compact subgroup. Proof. Let H be a torsion subgroup of a connected Lie group G and let § be its Lie algebra. Let Ad: G -» GL (§) be the adjoint representation of G and let § be the subalgebra of the associative algebra End(g) of all the endomorphisms of the linear space §, which is generated by Ad (77). As S is finitedimensional, there exists a finitely generated subgroup L of 77 such that the algebra S is generated by the subgroup Ad(L). By Lemma 1, Ad(L) is finite. As ker Ad is the center of G, we see that L is finite modulo its center. It follows then that L itself is finite. Let C be a maximal compact subgroup which contains L. We show that H < C. Indeed, let x E H. Since Ad {x) is contained in § which is generated by Ad(7) and Ad(L) leaves invariant the Lie Received by the editors March 21, 1975. AMS (MOS) subject classifications (1970). Primary 22C05, 22E20; Secondary 20F50. © American Mathematical Society 1976 424 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use on torsion subgroups of lie GROUPS 425 subalgebra G corresponding to C, we see that Ad(x) leaves G invariant. Thus x normalizes C. Since x is a torsion element, the maximality of C implies that x G C. Hence H < C. The proof of the following lemma may be found in [1, p. 286]. Lemma 3. Let G be a compact Lie group with Lie algebra § and let ad denote the adjoint representation of §. For each 0 < c < it, let Qc denote the totality of X G § such that all the eigenvalues of ad X have absolute value less than c. Let Uc = exp Qc. Then we have Uc = U~x, gUcg~x = Uc for all g G G, and Uc Uc> Q Uc+C., whenever 0 < c, c', c + c' < it. Now we are ready to prove the announced theorem. By Lemma 2, we may assume that G itself is compact. In [1] it is shown that if F is a finite subgroup of G, then F D i//3 is commutative. We claim that H fl £4/3 is commutative. In fact, if x, y G H fl £//3, then, by Lemma 1, {Ad(x), Ad(y)} generates a finite group, and hence {x,y} generates a finite subgroup of G and xy = yx follows. Thus H fl t/./3 generates an abelian subgroup. Let M denote the closure of H and let B denote the closure of the subgroup generated by H n Uv/3. Thus B is a closed abelian subgroup of the compact Lie group M. We claim that B fl H is a normal subgroup of H with finite index. In fact, the normality follows from the invariance of Uc under conjugation (Lemma 3). Let M0 denote the identity component of M. Since H is dense in M, we have M n t/./3 C H n t//3 C B, where, for any set A, A denotes the closure of A. Since M n Uy3 is a neighborhood of 1 in M, the subgroup generated by M n t//3 is open in M, and hence contains M0. Thus M0 < 5and we see that [A/: fi] < oo. Now [H: H D B] = [HB: B] < [M: B] < oo, proving that H n B is of finite index in H. It remains to show that the abelian normal subgroup H n B of H has index less that &(G). To do this, we proceed as in [1]. Let A = H n fi and let w = [//: ^4]. Let A, .4, ..., Am/i be the complete listing of the distinct cosets of A in H. Then the open sets hx U,6, ..., hm U,6 are pairwise disjoint. For otherwise hjU,6 n hjU,6 ¥= 0 for some ;' ¥= j which would imply that hj xh{ G U„/6Uw/6 C 2, m(A, Gw/6) = mn(U„/6), proving that [//: A] = m < A:(G). The following is immediate from Lemma 2. Corollary. A discrete torsion subgroup of a connected Lie group is finite. Using the main result, we prove the following: Corollary. Let G be a connected Lie group. Then G is a toroid if and only if G contains a dense torsion subgroup. Proof. Clearly a toroid contains a dense torsion subgroup. Assume, conversely, that G contains a dense torsion subgroup H. By the theorem, H contains an abelian normal subgroup A of finite index and A is compact by Lemma 2. Let xx A, ..., xmA be the complete listing of the cosets of A in H. Then G = U/lxxtA = Uj'lxXjA. Hence A is open and closed in G. Since G is connected, it follows that G = A proving that G is a toroid. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use