- Email: [email protected]
Regular representations of discrete groups
JOURNAL
OF FUNCTIONAL
ANALYSIS
11,
401-406 (1972)
Regular Representations MARTHA Department
of Mathematics,
of Discrete Groups SMITH
Rice University,
Houston, Texas 77001
Communicated by the Editors
Received November
4, 1971
Kaniuth [3] has given necessary and sufficient conditions for the regular representation of a discrete group to be type I or type II. We give a nonmeasure-theoretic proof of his results, plus additional information on the structure of the W* algebra generated by the regular representation.
Let pr denote the left regular representation of the discrete group G on H = L2( G). Let W(G) denote the closed linear span of pi(G) in the weak operator topology on g(H), the set of bounded operators on H. As described in [4], W(G) is the commuting algebra of the right regular representation of G and is a W* algebra of finite type. Consequently W(G) splits into type I and type II parts: there are mutually orthogonal central projections e = e(G), e, = e,(G), e2 = e,(G),..., in W(G) whose least upper bound is the identity, and such that eW(G) is type II and e, W(G) is type I, . More explicitly, e,W(G) is isomorphic to the n x n matrix ring over its center. Let G be any discrete group. Let d denote the subgroup consisting of all elements of G having only finitely many conjugates in G. Let d’ denote its commutator subgroup. E. Kaniuth [3] has proved the following two theorems. THEOREM 1. If G is any discrete group, then W(G) is of type II if and only ;f either
(i) (ii)
[G: d] = CO,or [G : A] < COand A’ is infinite.
THEOREM 2. If G is a discrete group, then W(G) is type I if and only if G has an abelian subgroup of finite index.
Kaniuth’s proofs employed a direct integral decomposition in the 401 Copyright Au rights
6 1972 by Academic Press, Inc. of reproduction in any form reserved.
SMITH
402
case when G is countable, and a reduction to the countable case. We will give proofs of Theorems 1 and 2 which are independent of direct integral theory. First, we shall introduce some convenient notation and list some useful facts of an algebraic nature. Let 7 tL2(G) be the characteristic function of the identity of G. As in [7], we may express any u E W(G) as xotC (a~) (cr) pi(o), in the sense of weak operator convergence. We shall identify 0 E G with its image p!(a) E *g(H), so that we shall write for a E W(G), a = 1 a,~, where a, = (a~) (u). In particular, u---f a, is a square-summable function. With this notation, multiplication in W(G) is just convolution. Two elements C u,u and C b,a of W(G) are equal if and only if a, = b, for every (Tt G. We define the support of a = 1 a,0 to be the set {u E G ~a, # 01. If a is central in W(G), then its support lies in d. If H is a subgroup of G, then W(H) may be embedded in W(G) as the IV* subalgebra of W(G) consisting of all elements whose support lies in H. G is called an FC group (for finite class) if G = d. If G is an FC group, then its commutator subgroup G’ is locally finite [8]. The following lemma is implied in [5, Theorem 9.111. For the sake of completeness, we include its proof here. An algebra A over a field F is said to satisfy a polynomial identity if there is a nonzero variables x1 ,..., x, polynomial f(~r ,..., x,) in the noncommuting for some n, such thatf(a, ,..., a,) = 0 for any choice of a, ,..., a, in A. LEMMA 1. polynomial
If W is a W* algebra of jkite identity, then W is of type I.
type
which
satisJies a
Proof. A matrix algebra of degree n over a field cannot satisfy a polynomial identity of degree less than 2n [l, Lemma 6.3.11. Thus W cannot contain a subalgebra isomorphic to C, for n > d/2, where d is the degree of some polynomial identity satisfied by W. But as in [lo], a type II W* algebra contains a subalgebra isomorphic to C, for any n. The following lemma is the key step in our proofs of Theorems 1 and 2. If H and K are subgroups of G, let (H, K) denote the subgroup of G generated by {h-lh-lhh 1 h E H, h E K}.
LEMMA 2. Suppose G is an FC group, H is a subgroup of G, and e is a central projection in W(G) such that both e W(H) [considering W(H) us a subalgebra of W(G)] and e W( G) are type I, . If C is the centralizer of H in G, then e(rru -17-l - 1) = 0 for every u E C, T E G. Consequently, (C, G) is finite. Proof.
Let p be an irreducible
representation
of eW(G).
Since
REGULAR
REPRESENTATIONS
403
eW(G) is a C* algebra isomorphic to the ring of n x n matrices over its center, we may assume that p maps eW(G) onto C, , the ring of n x n matrices over the complex field. Restricting p to eW(H), since p(e) # 0, deVf-0) = C, b ecause eW(H) is also a full n X n matrix ring. If (I E C, eu centralizes eW(H), so p(ea) centralizes p(eW(H)) = C, . Thus p(eo) commutes with p(eT) for every T E G, so p(e(oT - TU)) = 0. Since eW(G) h as a complete set of irreducible representations, e(aT - 70) = 0 and hence e(uTu-lr-l - 1) = 0, establishing the first statement to be proved. Now if we express e as Cgsc agg, this implies that a, = aoro-17m1for every u E C, T E G. Since C 1a, /a is finite, the set {u~u-+~ 1u E C, 7 E G} must be finite. Since this set generates (C, G) and is contained in the locally finite group G’, it follows that (C, G) is finite. We proceed to Theorem 1. Proof of Theorem 1. We shall show that W(G) has a type-1 summand if and only if both d’ and [G : d] are finite. Suppose W(G) h as a type-1 summand. By Mautner’s theorem [4, lo], [G : d] is finite. Furthermore, the matrix algebra e,W(G), and hence also its subalgebra e,W(d), satisfies a polynomial identity [l]. Therefore the W* algebra e,W(d) is type I by Lemma 2. But e, is also a central projection in W(d), so we have shown that W(L3) has a type I summand. Thus it is enough to show that if G is an FC group and W(G) has a type-1 part, then G’ is finite. So suppose G is an FC group and e, = e,(G) # 0. Let p, and p, be multilinear polynomials in dl and d, variables, respectively, such that for any full ring of m x m matrices R over a commutative ring, R satisfies p, if and only if m < n, and R satisfies p, if and only if m < n - 1 [l, Lemma 6.3.21. Then e,lV(G) satiesfies p, but not p, . Since multiplication in W(G) is weak operator continuous separately in each variable and since W(G) is generated as a IV* algebra by elements of G, there must exist ur ,..., ad, E G such that p2(emol ,..., enud,) f 0. Let H be the (normal) subgroup of G generated by uI ,..., ad, and all their (finitely many) conjugates in G. Then e,W(H) satisfies p, , so by Lemma 1, e,W(H) is type-I, and in fact by the property of p, , has no summands of type-I, for m > n. Also, e,W(H) does not satisfy p, . Therefore e,W(H) has a type I, summand. Let eW(H) be the maximal type I, summand of e,W(H) (cf. [6]). Since His normal in G, for any u E G the inner automorphism determined by u induces an algebra automorphism of e,@‘(H) and hence of its maximal type-I, summand eW(H). Consequently a-leuW(H) = eW(H), so by uniqueness of e, u-lea = e. Thus
SMITH
404
e is central in W(G), eW(H) is type I,, and eW(G) is clearly also Lemma 2, (C, G) must be finite where C is the centralizer of H in G. In particular the commutator subgroup C’ of C is finite. Since H is finitely generated and G is an FC group, C is of finite index in G. Thus G has a subgroup of finite index whose commutator subgroup is finite. This implies [9, Lemma 4.11 that G’ is finite. Conversely, suppose that [G : d] and d’ are both finite. Let f = (l/l A’ I) COCd’ (3. Then f is a central projection in W(G) and fW(0) is commutative. If (0i ,..., unL} is a complete set of coset representatives of d in G, then fW(G) is a direct sum of the free left f W(d) submodules f W(d) u1 ,..., f W(d) u,, . Thus by representing fW(G) by right multiplication on itself, we may embed f W(G) in the m x m matrices over fW(0). H ence f W(G) satisfies a polynomial identity [I]. By Lemma 2, it is then a type-1 direct summand of W(G). The next lemma gives information about the projections e(G) and e,(G) themselves, and provides the link between Theorems 1 and 2.
type 1,. Applying
LEMMA 3. support in G’.
If
G is any discrete group, then e,(G) and e(G) have.
Proof. Let y be a complex linear character of G. y induces a *-automorphism of W(G) by mapping C, a,g to C, u&d g. Since may be defined uniquely in terms of the algebraic e(G) and e,(G) properties and involution on W(G) [6], they are fixed by all *-automorphisms of W(G). In particular, they are fixed by the automorphism induced by y. This implies that r(g) = 1 for every g in the support of e or in the support of e, . Since y is any linear character and since G/G’ possesses a complete set of complex linear characters, this forces g E G’. COROLLARY. If f(G) finitely many integers n.
h as a type-1 summand, then e,(G) # 0 for only
Proof. First suppose G is an FC group. By Lemma 3, each e, E W(G’). By Theorem 1, G’ is finite. Therefore W(G’) is finite dimensional and so can contain only finitely many nonzero, mutually orthogonal idempotents. Hence only finitely many of the e, are nonzero. In the general case, it follows as in the proof of Theorem 1 that e,(G) W(d) is type I for each n. Therefore letting e = e(G), (1 - e) W(d) is type I. Since the corollary holds for W(d), (1 - e) W(d) is th e sum of finitely many type-I, parts. Let C be the
REGULAR
405
REPRESENTATIONS
ring direct sum of the centers of the maximal type-I, parts of (1 - e) W(d), as n varies. Then (1 - e) W(d) may be naturally embedded in the n x n ring C, over the commutative ring C, for a suitably large integer n. Now (1 - e) W(G) is a direct sum of the free left (1 T e) W(d) submodules (1 - e) W(d) ui ,..., (1 - e) W(d) u, , where (ui ,..., CJ,} is a complete set of coset representatives for d in G. ring Hence (1 - e) W(G) may be embedded in the matrix [(I - e) W(d)], . Thus (1 - e) W(G) C [( 1 - e) IV@)], C (C,), ‘v C,, . It follows that (1 - e) IV(G) sat is fi es a polynomial the corollary. We turn now to Theorem 2.
identity,
proving
Proof of Theorem 2. Suppose that G has an abelian subgroup of finite index n. Then W(A) is commutative and
A
W(G) = W(A) u1 @ .-a @ W(A) u, as a left W(A) module, where {a, ,..., u,} is a complete set of coset representatives for A in G. As in the proof of Theorem 1, this implies that IV(G) satisfies a polynomial identity and hence is type I. Conversely, suppose W(G) is type I. As in Theorem 1, [G : d] < 00 and W(d) is also of type I. Thus it will be enough to prove the theorem for G an FC group. So suppose G is an FC group. By Theorem 1, G’ is finite. Therefore we can find a maximal set fi ,..., fk of nonzero projections in W(G’) which are minimal with respect to being central in W(G). Then fi + a** + fk = 1. By Lemma 3, e,(G) E W(G’) for each n. The e,(G) are orthogonal and add up to the identity, so by minimality of fi , fie,(G) is nonzero for precisely one n, say ni , and fie,,(G) = fi . Thus fi W(G) is of type ni . As in the proof of Theorem 1, choose a finitely generated subgroup Hi of G such that fiW(Hi) has a type-I,, part. We may assume that G’ C Hi , so that fi E W(HJ. As in Theorem 1, e,(H,) is central in W(G) for each Y. By Lemma 3,
e,(H,) E W(H,‘) C W(G,‘), so as above we see that fi = fie,(Hi) for some r, so that fiW(Hi) is type I,. But fiW(H,) was chosen to have a type-Iai part, forcing r = lzi . Thus both fiW(Hi) and f,W(G) are of type I,* . By Lemma 2, letting C, be the centralizer of Hi in G, fi(um-kl - 1) = 0 for every u E Ci , 7 E G. Let C = ($=, Ci . Then for u E C and 7 E G,
406
SMITH
l - 1) = 0 for every i, forcing UTU-~T-~ = 1. Thus C is contained in the center of G. Since each Hi is finitely generated and G is an FC group, each Ci is of finite index in G. Therefore C also has finite index in G and is the desired abelian subgroup of finite index.
fi( cm-+
REFERENCES 1. I. N. HERSTEIN, “Non-Commutative Rings,” Carus Math. Monographs No. 15, M.A.A., 1968. 2. N. JACOBSON, “Structure of Rings,” A.M.S. Colloquium Publications, Vol. 27, American Mathematical Society, Providence, RI, 1956. 3. E. KANIUTH, Der Typ der regularen Darstellung diskreter Gruppen, Math. Ann. 182 (1969), 334-339. 4. I. KAPLANSKY, Group algebras in the large, Tohoku Math. J. 3 (1951), 249-256. The structure of certain operator algebras, Trans. Amer. Math. 5. I. KAPLANSKY, Sot. 70 (1951), 219-255. 6. I. KAPLANSKY, “Rings of Operators,” Benjamin, New York, 1968. 7. G. MACKEY, “The Theory of Group Representations,” University of Chicago Lecture Notes, Chicago, Ill., 1955. 8. B. H. NEUMANN, Groups with finite classes of conjugate elements, PYOC.London Math. Sot. 1 (1951), 178-187. 9. B. H. NEUMANN, Groups with finite classes of conjugate subgroups, Math. Z. 63 (1955), 76-96. 10. M. SMITH, Group algebras, J. Algebra 18 (1971), 477-499.
OF FUNCTIONAL
ANALYSIS
11,
401-406 (1972)
Regular Representations MARTHA Department
of Mathematics,
of Discrete Groups SMITH
Rice University,
Houston, Texas 77001
Communicated by the Editors
Received November
4, 1971
Kaniuth [3] has given necessary and sufficient conditions for the regular representation of a discrete group to be type I or type II. We give a nonmeasure-theoretic proof of his results, plus additional information on the structure of the W* algebra generated by the regular representation.
Let pr denote the left regular representation of the discrete group G on H = L2( G). Let W(G) denote the closed linear span of pi(G) in the weak operator topology on g(H), the set of bounded operators on H. As described in [4], W(G) is the commuting algebra of the right regular representation of G and is a W* algebra of finite type. Consequently W(G) splits into type I and type II parts: there are mutually orthogonal central projections e = e(G), e, = e,(G), e2 = e,(G),..., in W(G) whose least upper bound is the identity, and such that eW(G) is type II and e, W(G) is type I, . More explicitly, e,W(G) is isomorphic to the n x n matrix ring over its center. Let G be any discrete group. Let d denote the subgroup consisting of all elements of G having only finitely many conjugates in G. Let d’ denote its commutator subgroup. E. Kaniuth [3] has proved the following two theorems. THEOREM 1. If G is any discrete group, then W(G) is of type II if and only ;f either
(i) (ii)
[G: d] = CO,or [G : A] < COand A’ is infinite.
THEOREM 2. If G is a discrete group, then W(G) is type I if and only if G has an abelian subgroup of finite index.
Kaniuth’s proofs employed a direct integral decomposition in the 401 Copyright Au rights
6 1972 by Academic Press, Inc. of reproduction in any form reserved.
SMITH
402
case when G is countable, and a reduction to the countable case. We will give proofs of Theorems 1 and 2 which are independent of direct integral theory. First, we shall introduce some convenient notation and list some useful facts of an algebraic nature. Let 7 tL2(G) be the characteristic function of the identity of G. As in [7], we may express any u E W(G) as xotC (a~) (cr) pi(o), in the sense of weak operator convergence. We shall identify 0 E G with its image p!(a) E *g(H), so that we shall write for a E W(G), a = 1 a,~, where a, = (a~) (u). In particular, u---f a, is a square-summable function. With this notation, multiplication in W(G) is just convolution. Two elements C u,u and C b,a of W(G) are equal if and only if a, = b, for every (Tt G. We define the support of a = 1 a,0 to be the set {u E G ~a, # 01. If a is central in W(G), then its support lies in d. If H is a subgroup of G, then W(H) may be embedded in W(G) as the IV* subalgebra of W(G) consisting of all elements whose support lies in H. G is called an FC group (for finite class) if G = d. If G is an FC group, then its commutator subgroup G’ is locally finite [8]. The following lemma is implied in [5, Theorem 9.111. For the sake of completeness, we include its proof here. An algebra A over a field F is said to satisfy a polynomial identity if there is a nonzero variables x1 ,..., x, polynomial f(~r ,..., x,) in the noncommuting for some n, such thatf(a, ,..., a,) = 0 for any choice of a, ,..., a, in A. LEMMA 1. polynomial
If W is a W* algebra of jkite identity, then W is of type I.
type
which
satisJies a
Proof. A matrix algebra of degree n over a field cannot satisfy a polynomial identity of degree less than 2n [l, Lemma 6.3.11. Thus W cannot contain a subalgebra isomorphic to C, for n > d/2, where d is the degree of some polynomial identity satisfied by W. But as in [lo], a type II W* algebra contains a subalgebra isomorphic to C, for any n. The following lemma is the key step in our proofs of Theorems 1 and 2. If H and K are subgroups of G, let (H, K) denote the subgroup of G generated by {h-lh-lhh 1 h E H, h E K}.
LEMMA 2. Suppose G is an FC group, H is a subgroup of G, and e is a central projection in W(G) such that both e W(H) [considering W(H) us a subalgebra of W(G)] and e W( G) are type I, . If C is the centralizer of H in G, then e(rru -17-l - 1) = 0 for every u E C, T E G. Consequently, (C, G) is finite. Proof.
Let p be an irreducible
representation
of eW(G).
Since
REGULAR
REPRESENTATIONS
403
eW(G) is a C* algebra isomorphic to the ring of n x n matrices over its center, we may assume that p maps eW(G) onto C, , the ring of n x n matrices over the complex field. Restricting p to eW(H), since p(e) # 0, deVf-0) = C, b ecause eW(H) is also a full n X n matrix ring. If (I E C, eu centralizes eW(H), so p(ea) centralizes p(eW(H)) = C, . Thus p(eo) commutes with p(eT) for every T E G, so p(e(oT - TU)) = 0. Since eW(G) h as a complete set of irreducible representations, e(aT - 70) = 0 and hence e(uTu-lr-l - 1) = 0, establishing the first statement to be proved. Now if we express e as Cgsc agg, this implies that a, = aoro-17m1for every u E C, T E G. Since C 1a, /a is finite, the set {u~u-+~ 1u E C, 7 E G} must be finite. Since this set generates (C, G) and is contained in the locally finite group G’, it follows that (C, G) is finite. We proceed to Theorem 1. Proof of Theorem 1. We shall show that W(G) has a type-1 summand if and only if both d’ and [G : d] are finite. Suppose W(G) h as a type-1 summand. By Mautner’s theorem [4, lo], [G : d] is finite. Furthermore, the matrix algebra e,W(G), and hence also its subalgebra e,W(d), satisfies a polynomial identity [l]. Therefore the W* algebra e,W(d) is type I by Lemma 2. But e, is also a central projection in W(d), so we have shown that W(L3) has a type I summand. Thus it is enough to show that if G is an FC group and W(G) has a type-1 part, then G’ is finite. So suppose G is an FC group and e, = e,(G) # 0. Let p, and p, be multilinear polynomials in dl and d, variables, respectively, such that for any full ring of m x m matrices R over a commutative ring, R satisfies p, if and only if m < n, and R satisfies p, if and only if m < n - 1 [l, Lemma 6.3.21. Then e,lV(G) satiesfies p, but not p, . Since multiplication in W(G) is weak operator continuous separately in each variable and since W(G) is generated as a IV* algebra by elements of G, there must exist ur ,..., ad, E G such that p2(emol ,..., enud,) f 0. Let H be the (normal) subgroup of G generated by uI ,..., ad, and all their (finitely many) conjugates in G. Then e,W(H) satisfies p, , so by Lemma 1, e,W(H) is type-I, and in fact by the property of p, , has no summands of type-I, for m > n. Also, e,W(H) does not satisfy p, . Therefore e,W(H) has a type I, summand. Let eW(H) be the maximal type I, summand of e,W(H) (cf. [6]). Since His normal in G, for any u E G the inner automorphism determined by u induces an algebra automorphism of e,@‘(H) and hence of its maximal type-I, summand eW(H). Consequently a-leuW(H) = eW(H), so by uniqueness of e, u-lea = e. Thus
SMITH
404
e is central in W(G), eW(H) is type I,, and eW(G) is clearly also Lemma 2, (C, G) must be finite where C is the centralizer of H in G. In particular the commutator subgroup C’ of C is finite. Since H is finitely generated and G is an FC group, C is of finite index in G. Thus G has a subgroup of finite index whose commutator subgroup is finite. This implies [9, Lemma 4.11 that G’ is finite. Conversely, suppose that [G : d] and d’ are both finite. Let f = (l/l A’ I) COCd’ (3. Then f is a central projection in W(G) and fW(0) is commutative. If (0i ,..., unL} is a complete set of coset representatives of d in G, then fW(G) is a direct sum of the free left f W(d) submodules f W(d) u1 ,..., f W(d) u,, . Thus by representing fW(G) by right multiplication on itself, we may embed f W(G) in the m x m matrices over fW(0). H ence f W(G) satisfies a polynomial identity [I]. By Lemma 2, it is then a type-1 direct summand of W(G). The next lemma gives information about the projections e(G) and e,(G) themselves, and provides the link between Theorems 1 and 2.
type 1,. Applying
LEMMA 3. support in G’.
If
G is any discrete group, then e,(G) and e(G) have.
Proof. Let y be a complex linear character of G. y induces a *-automorphism of W(G) by mapping C, a,g to C, u&d g. Since may be defined uniquely in terms of the algebraic e(G) and e,(G) properties and involution on W(G) [6], they are fixed by all *-automorphisms of W(G). In particular, they are fixed by the automorphism induced by y. This implies that r(g) = 1 for every g in the support of e or in the support of e, . Since y is any linear character and since G/G’ possesses a complete set of complex linear characters, this forces g E G’. COROLLARY. If f(G) finitely many integers n.
h as a type-1 summand, then e,(G) # 0 for only
Proof. First suppose G is an FC group. By Lemma 3, each e, E W(G’). By Theorem 1, G’ is finite. Therefore W(G’) is finite dimensional and so can contain only finitely many nonzero, mutually orthogonal idempotents. Hence only finitely many of the e, are nonzero. In the general case, it follows as in the proof of Theorem 1 that e,(G) W(d) is type I for each n. Therefore letting e = e(G), (1 - e) W(d) is type I. Since the corollary holds for W(d), (1 - e) W(d) is th e sum of finitely many type-I, parts. Let C be the
REGULAR
405
REPRESENTATIONS
ring direct sum of the centers of the maximal type-I, parts of (1 - e) W(d), as n varies. Then (1 - e) W(d) may be naturally embedded in the n x n ring C, over the commutative ring C, for a suitably large integer n. Now (1 - e) W(G) is a direct sum of the free left (1 T e) W(d) submodules (1 - e) W(d) ui ,..., (1 - e) W(d) u, , where (ui ,..., CJ,} is a complete set of coset representatives for d in G. ring Hence (1 - e) W(G) may be embedded in the matrix [(I - e) W(d)], . Thus (1 - e) W(G) C [( 1 - e) IV@)], C (C,), ‘v C,, . It follows that (1 - e) IV(G) sat is fi es a polynomial the corollary. We turn now to Theorem 2.
identity,
proving
Proof of Theorem 2. Suppose that G has an abelian subgroup of finite index n. Then W(A) is commutative and
A
W(G) = W(A) u1 @ .-a @ W(A) u, as a left W(A) module, where {a, ,..., u,} is a complete set of coset representatives for A in G. As in the proof of Theorem 1, this implies that IV(G) satisfies a polynomial identity and hence is type I. Conversely, suppose W(G) is type I. As in Theorem 1, [G : d] < 00 and W(d) is also of type I. Thus it will be enough to prove the theorem for G an FC group. So suppose G is an FC group. By Theorem 1, G’ is finite. Therefore we can find a maximal set fi ,..., fk of nonzero projections in W(G’) which are minimal with respect to being central in W(G). Then fi + a** + fk = 1. By Lemma 3, e,(G) E W(G’) for each n. The e,(G) are orthogonal and add up to the identity, so by minimality of fi , fie,(G) is nonzero for precisely one n, say ni , and fie,,(G) = fi . Thus fi W(G) is of type ni . As in the proof of Theorem 1, choose a finitely generated subgroup Hi of G such that fiW(Hi) has a type-I,, part. We may assume that G’ C Hi , so that fi E W(HJ. As in Theorem 1, e,(H,) is central in W(G) for each Y. By Lemma 3,
e,(H,) E W(H,‘) C W(G,‘), so as above we see that fi = fie,(Hi) for some r, so that fiW(Hi) is type I,. But fiW(H,) was chosen to have a type-Iai part, forcing r = lzi . Thus both fiW(Hi) and f,W(G) are of type I,* . By Lemma 2, letting C, be the centralizer of Hi in G, fi(um-kl - 1) = 0 for every u E Ci , 7 E G. Let C = ($=, Ci . Then for u E C and 7 E G,
406
SMITH
l - 1) = 0 for every i, forcing UTU-~T-~ = 1. Thus C is contained in the center of G. Since each Hi is finitely generated and G is an FC group, each Ci is of finite index in G. Therefore C also has finite index in G and is the desired abelian subgroup of finite index.
fi( cm-+
REFERENCES 1. I. N. HERSTEIN, “Non-Commutative Rings,” Carus Math. Monographs No. 15, M.A.A., 1968. 2. N. JACOBSON, “Structure of Rings,” A.M.S. Colloquium Publications, Vol. 27, American Mathematical Society, Providence, RI, 1956. 3. E. KANIUTH, Der Typ der regularen Darstellung diskreter Gruppen, Math. Ann. 182 (1969), 334-339. 4. I. KAPLANSKY, Group algebras in the large, Tohoku Math. J. 3 (1951), 249-256. The structure of certain operator algebras, Trans. Amer. Math. 5. I. KAPLANSKY, Sot. 70 (1951), 219-255. 6. I. KAPLANSKY, “Rings of Operators,” Benjamin, New York, 1968. 7. G. MACKEY, “The Theory of Group Representations,” University of Chicago Lecture Notes, Chicago, Ill., 1955. 8. B. H. NEUMANN, Groups with finite classes of conjugate elements, PYOC.London Math. Sot. 1 (1951), 178-187. 9. B. H. NEUMANN, Groups with finite classes of conjugate subgroups, Math. Z. 63 (1955), 76-96. 10. M. SMITH, Group algebras, J. Algebra 18 (1971), 477-499.