Pacific Journal ofMathematics

REMARKS ON A THEOREM OF L. GREENBERG ON THEMODULAR GROUP

A. W. MASON AND WALTER WILSON STOTHERS

Vol. 78, No. 2 April 1978

PACIFIC JOURNAL OF MATHEMATICSVol. 78, No. 2, 1978

REMARKS ON A THEOREM OF L. GREENBERGON THE MODULAR GROUP

A. W. MASON AND W. W. STOTHERS

Introduction* For integers a and 6, each greater than 1, letT(a, b) be the free product of cyclic groups of orders a and b. ThenT(a, b) has presentation

<X, Y: Xa = Yb - 1> .

Suppose that G<\T(a, b). If XYG has finite order in Γ(α, b)/G,then the order is the level of 6?, denoted by n(G). We put U — XY.When Cr has finite index μ{G), then n(G) is defined, and dividesμ{G). In such a case, t(G) — μ(G)/n(G) is the parabolic class numberof (?. These definitions agree with the usual ones for Γ(2, 3), theclassical modular group.

For 3Γ(2, 3), Newman [7] raised the question of whether therewere infinitely many normal subgroups with a given parabolic classnumber. In [3], L. Greenberg showed that this was not possible byproving that, for t > 1,

Here, as later, we write μ, t for μ(G), t(G) when the group is clearfrom the context.

Mason [5] improved this to

(1) μ ^ t* .

This was also proved by Accola [1]. Implicit in his proof is a proofthat (1) holds when a and b are distinct primes.

Here, we show that, when a and b are coprime, there is aconstant c(α, b) such that, for t > 1,

( 2 ) μ£ c{a, b)t\t - 1) .

The constant is 1 when a and b are distinct primes, e.g., for themodular group. There is no corresponding result when a and b arenot coprime.

We give examples to show that we can have equality in (2), butonly a finite number of times for given a and δ. Finally, we obtaina better result for large t.

The referee has drawn our attention to a paper of MorrisNewman, '2-generator groups and parabolic class numbers', Proc.Amer. Math. Soc, 31 (1972), 51-53, which contains the weaker result

385

386 A. W. MASON AND W. W. STOTHERS

μ ^ ab ta+1 .

1* Preliminary results •

PROPOSITION 1.1. Suppose that K is a finite group, and thatH = (U) is a cyclic subgroup with order k<\K\9 and withΓlveκ VHV~ι = {1}. If K= <X, U), then HnXHX1 = {1}, and thecosets H, XH, UXH, •-, U^XH are distinct.

Proof. Let E = HftXHX1. As H is cyclic, so is E, and henceE<\K. Thus, E= {1}. The last clause follows at once.

We observe that, in the situation described in 1.1, K acts as atransitive permutation group on the cosets on H.

PROPOSITION 1.2. Suppose that a and b are coprime, and thatG <\ T(a, b) with index μ{G). Then there is a normal subgroup 6?*of T{a, b) with G 6?* and such that

( i ) t(G*) = t(G), =t say,(ii) if t > 1, then μ(fi*) ^ t(t - 1),(iii) G*/G is central in T(a, b)/G.

Proof Let D = (U, G), then | Γ(α, b): D\ = t{G).Let 6?* = Γ\veτia,b) VDV~\ so that 6? 6?* < Γ(α, 6). As /)/(? is

cyclic, G* = (Uk, G), for a least positive integer k. As /> = <ί7, 6?*>,n(6?*) - k, so (i) holds.

If t > 1, then Z> is proper. From 1.1 applied to K = Γ(α, 6)/6?>iί

and JζΓ- Z>/6?*, it follows that t ^ k + 1, so (ii) holds.For VeT(a,b), let [F] denote the corresponding element of

Aut(G*/G). Then [Xf = [Yf = 1, and [X][Y] = [U] = I, so (iii)holds. (Cf. Lemma 3 of [3].)

COROLLARY 1.3. With the notation of 1.2,( i ) if X or Ye Gf then t{G) - 1,(ii) if t(G) = 1, then μ(G)\ab.

Proof. We observe that t(G) = 1 if and only if D = T(a, b).If X or YeG, then D = Γ(α, 6), and (i) holds.If t(G) = 1, then G* = T(a, b). By 1.2 (iii), Γ(α, 6)/G is abelian,

so that (ii) holds.

For integers a and b with 1/α + 1/6 < 1, there is a Fuchsiangroup of the first kind isomorphic to T(a, b). The details can befound in [4], We write T(a, b) for the Fuchsian group as well asfor its abstract counterpart, taking the isomorphism so that U cor-

REMARKS ON A THEOREM OF L. GREENBERG ON THE MODULAR GROUP 387

responds to a mapping ω t-> ω + a, with a > 0.Also from [4], a subgroup of finite index in T(a, b) has a pre-

sentation

K, , Er, Plf - , P,, Alf Bu -. , A,, £,: JE, elliptic,(3)

In this presentation, P^ , P t are parabolic, i.e., each is T(a9b)-conjugate to a power of U. The amplitude of a parabolic elementis the exponent of U. We can choose the presentation with each Pt

operating anti-clockwise. This will be described as the standardpresentation.

PROPOSITION 1.4. In the standard presentation for a subgroupof T(a, b), each parabolic generator has negative amplitude.

The proof is exactly that given for Theorem 1 in [5].

2* The inequality (2)* We write n' for the largest properdivisor of a positive integer n.

THEOREM 2.1. Suppose that a and b are coprime, and thatG <| T(a, b) with index μ. If t > 1, then

μ ^ a'Vt\t - 1) .

Proof. By 1.2, there is a subgroup 6r* of index kt, withk ^ t — 1, and with properties (i), (ii), and (iii).

By a standard argument on Fuchsian groups, a finite subgroupof G* is Γ(α, &)-conjugate to a subgroup of (X) or of (Y). As G*is normal, we can divide such subgroups into those of order β, withβ\a, and those of order /, with f\b. As t > 1, 1.3 applies, so e ^ a'and / £ b'.

In the standard presentation of (?*, each elliptic generator hasorder e or /. Using 1.4 and the normality of 6?*, each parabolicgenerator has amplitude — k. As G*/G is central, the efth power ofthe relation in (3) yields

Thus, n(G)\efkt. The result follows.

COROLLARY 2.2. If a and b are distinct primes, and G is asin the theorem, then μ ^ t\t — 1).

The inequality (2) is best possible, as the following examplesshow:

388 A. W. MASON AND W. W. STOTHERS

(a) in the notation of [7], (Γ2)' and GZΛ are normal subgroups ofΓ(2, 3) and have, respectively, μ = 18, t = 3 and μ = 48, t = 4,

(b) in Γ(3, 4), 6? = <X, ΓXΓ3, Γ2> has index 2 and is isomorphic tothe free product C?C?C9. The product of the generators is ( J 7 ) 2 , son(G) = 2 and ί(6?) - 1. Thus, Gf <\ Γ(3, 4) and has μ = 36, t = 3.Example (b) is analogous to the first example in (a). As we shallsee in § 4, there are no further examples for Γ(2, 3).

3* Non-coprime cases* In this section, we suppose that a andb have g.c.d. (α, &) = c£, with d > 1. We produce an infinite collec-tion of subgroups of Γ(α, 6), each with parabolic class number d.Intersecting these with other normal subgroups, we see that therecan be no inequality of the form μ^f(t).

We begin by considering T(d, d). Let H = ( 1 7 , T(d, d)'). Thenthe Reidemeister-Schreier method shows that H, which is normal inT(d, d), has presentation

Ar = Y'XY1-', r = 0, 1, , d - 1: R Ar = 1r=0

Then 27 is free on the first d — 1 of these generators. Further,T(d,d) = (Y,H), so that \T(d, d): H\ = d. Finally, Ao = XY, sothat tt(U) = 1 and t(H) = d.

As (Z > 1, Dirichlet's theorem states that there are an infinitenumber of primes congruent to 1 modulo d. For any such prime,there is an integer e with oτάp(e) = d. We define H(p, e) by

H(p, β) - # ' <(A)P, 5, = ( Λ ^ r A, r = 1, , d - 2> .

This is invariant under XY. Also, for r = 0, •••, c£ — 2, YArY~ι =Ar+1, so that, for r = 1, , d - 3, Γ ^ Γ " 1 = £ r + 1 . Finally, YBd_2Y-1

can be expressed, modulo ZΓ, in terms of the Br and (A0)p. Thus,

, e) is normal. (The proof for d = 2 is simpler.)Since Ao = X7, £Γ(p, e) has level p. Also, \H: H(p9 e)\ = p and

= d, so that ί(J7(p, ej) = d.To obtain subgroups of Γ(α, 6), we observe that, if N is the

normal closure of {X\ Yd) in Γ(α, 6), then T(d, d) = T(a, b)/N. ThenT(a, b) has subgroups with level p and parabolic class number d foran infinite set of p.

4. Frobenius factor groups* Throughout this section, we shallassume that a and b are coprime, and that t > 1. We adopt thenotation of 1.2, and of 1.1 with K = Γ(α, &)/(? and H = 2>/G, andwrite X* and ZP for the corresponding groups with G replaced by 6r*.We regard K (resp. if*) as a transitive group on the cosets of H(resp. H*). Our results describe the situation where there is equalityin 1.2 (ii).

REMARKS ON A THEOREM OF L. GREENBERG ON THE MODULAR GROUP 389

THEOREM 4.1. If k — t ~ 1, then K* is primitive.

Proof. In this case, 1.1 shows that ϋΓ* is doubly transitive.

THEOREM 4.2. Suppose that K* is primitive. Then K* is aFrobenius group with kernel (K*Y, which is elementary dbelian.Further, there is a prime p with t — pn = 1 (mod ft), and n — ordk(p).Finally, k\ab.

Proof. Note that H*φK*, since ί > l . Suppose that Ve # * - # * .As K* is primitive, JBΓ* is maximal, so that If* = (V, Z76r*>. By1.1, if VH*ΦWH*, then VH^V'1 n WH*W~ι - {1}. Thus, JRΓ* isFrobenius with kernel JV where |JV| = t. By [10, p. 30], JV is ele-mentary abelian, so that t = pn.

Let ΛΓ< # * with M^N. If 1 < Jf < JV, then JET * < H*M<K*which contradicts the maximality of JEΓ*. Thus Jf = {1} or JV. AsK*IN^ H*f (K*y £ JV, so that (K*)' = JV.

By the general theory of Frobenius groups, k = | J9Γ* | dividesp* — 1 = I JV| — 1. As U acts irreducibly on JV, [2, p. 212] showsthat, if ω is a primitive Λ-th root of unity over GF(p), then ωp\ i — 1,w — 1, are distinct. Hence ^ = ordfc(p).

For the last part, we observe that fc = |Γ(α, δ): (Γ(α, δ))' G*|which divides |Γ(α, δ): (Γ(α, δ))'| = ab.

Combining 4.1 and 4.2, we obtain

COROLLARY 4.3. If k = t — 1, £&ew (ί — l))αδ. For fixed a andδ, ίfeere are finitely many normal subgroups with equality in (2).

COROLLARY 4.4. If l/2ί k <t — 1, If* is imprimitive.

Proof. This follows at once, since we cannot have ί = l (mod &).

We note that, if G <| Γ(2, 3) has genus 1 and t > 4, then 6?* = Gand ft = 6, see [7]. By [9, p. 181], if 6|ί - 1, then K* is Frobenius,so that K* Frobenius does not imply ϋΓ* primitive. Further, for aprime p > 3, there is a primitive K * with k — 6, t — pn, wherew = ord3(p). These subgroups have ft < ί/2 in general.

Theorem 4.2 has a converse, as we shall now show. Let p be aprime and ft an integer prime to p. Let w = oτάk(p), and write Sfor the cyclic subgroup of order ft in GF(pn)*. Let F = {(x, y):xeS,y e GF{pn)}> and define multiplication on F by

(%, y)*{u, v) = (αw, yu + v) .

390 A. W. MASON AND W. W. STOTHERS

PROPOSITION 4.5. F is a primitive Frobenius group on pn

symbols. The kernel is Ff, which is elementary abelian, and thecomplement C is cyclic of order k.

Proof. It is clear that F is Frobenius with kernel N = {(1, y):y 6 GF(pn)} and complement C = {(x, 0):xeS}.

If 1 < M < N with M<\F, then, by [9, p. 183], F/M is Fro-benius with kernel N/M. Then k = \C\ divides \N: M\ - l = pm - 1,where 1 < m < n. This contradicts the definition of n. Hence,N=F'.

If C < M< F, then I i f I = prk, where 1 r < n. Then \Mf]N\ = pr

and, by [9, p. 183], M is Frobenius with kernel Mf]N. Hence,k\(pr — 1), again a contradiction. Thus, C is maximal and F isprimitive on the cosets of C.

PROPOSITION 4.6. With the above notation, suppose that k = e/,with (e, f) = 1. I%ew we /&α?;e,

( i ) if e,f> 1, then F = <x, #>, wiί/z, a; o/ order e, y of order f.(ii) if e = 1, ί/ w F = <«, 2/>, wiίfe x o/ orcZer p, 7/ of order k.

Proof, (i) As e,f\k, we can take xeC, yeCz, with zeF—C,with x of order e and 2/ of order / . Let M= (x, y). As [x, y] e N— {1},then I i f I = prk, where 1 r ^ w. By [9, p. 183], k\(pr - 1), sothat r — w and M — F.

(ii) We take cceiV, of order p and ?/ a generator of C Theresult follows as in (i).

Since the center of a Frobenius group is trivial,

LEMMA 4.7. If G <\ T(a, b) with K Frobenius, then G = 6?*.

LEMMA 4.8. Let G <J Γ(α, 6) with K a primitive Frobenius groupwith elementary p-abelian kernel and complement C cyclic of order k.Then, (i) C is conjugate to H, and

(ii) if n(G) is prime to a, then p\a.

Proof. By 4.7, G - <S*, so that H = H* and K = X*. Letif/6? - N be the kernel of K, and let |X| = p%&, so (p, k) = 1.

( i ) Since T(a, b)/M = C, n{M) - k. Hence (UG)kp = 1 in K.By [9, p. 182], either (E/6?)* - 1 or (UGy = 1. It follows that([/(?)* - 1 and so Nf)H = {1}. Thus, £Γ is a complement of N in ϋΓ.By [9, p. 186], H is a conjugate of C.

(ii) Since M< Γ(α, 6) and | Γ(α, δ): M\ = k, X e M. Thus, X* e Gand, if (α, p) = 1, then XeG which would imply that K is abelian.Thus, p\a.

REMARKS ON A THEOREM OF L. GREENBERG ON THE MODULAR GROUP 391

THEOREM 4,9. Given k > 1, a divisor of α&, and p prime to k(with p\a if (k, a) = 1, p\b if (fc, b) = 1), έ&ere is α subgroupG <J Γ(α, 6) wiίft If* primitive Frobenius of order pnk, wheren =

Proof. Let fc = e/, where e|α and / |6 , so that (e, /) = 1. Theresult follows from 4.5, 4.6, 4.7, and 4.8.

Thus, when ab + 1 is #% with p prime, there is a subgroup ofmaximal index with equality in 1.2 (ii). However, there is no cor-responding subgroup with equality in (2), as we now show.

THEOREM 4.10. (Γ(α, b))" has level ab.

Proof. It is clear that (Γ(α, δ))' is free, has level ab and para-bolic class number 1. The standard presentation shows that Uabe(T(a, b))". Since the level is a multiple of ab, the result follows.

This is proved for a = 2 and b = 3 in [8].

THEOREM 4.11. If k = ί - 1 = αδ, ίfcew G = G*.

Proo/. Assume that A; = ί - 1 = αδ, but that <? =£ 6?*. By 4.1and 4.2, K* is Frobenius with kernel of index k = ab. Also, t = pn, pprime. Let ifcΓ/G* be the kernel. Then | T(a, b): Mj = ab. AsT{a, b)/M is cyclic, M = T(a, b)f. Thus, 6?* is free, so that the proofof 2.1 shows that \G*\G\ = p% with s ^ 1. By 1.2 (iii), there is asubgroup L <] Γ(α, 6) with G^L^G* and |£?*: Z,| = p.

Let A=T(a,b)/M and P^MjL. By 4.10, M'L^G*, soP' - 6?*/X. Now, P f ^ Z(JP), and, since Γ(α, δ)/ΛΓ acts irreduciblyon M/G*, Pf = Z(P). The Frattini subgroup Φ(P) of P is the smal-lest normal subgroup with elementary abelian factor, [2, p. 174].Thus, P' = Z(P) = Φ(P), so that P is an extra-special p-group, [2,p. 183].

Let A — <α>, with a regarded as an element of order d of Aut(P).Then, in Aut(P/P'), ad == 1. Considering the action of A on M/G*, ahas order p% — 1 as an element of Aut(P/P'). Thus, d ~ pn — 1.

By [2, p. 213], pn - l |p ' + 1, with r ^ n/2. Thus, p% = 4, sothat ab — 3. As (α, 6) = 1, this is impossible. Hence G = 6?*.

COROLLARY 4.12. If G <\ Γ(2, 3) flfws equality in (2), ίΛβw ί = 3o?* 4.

Proo/. By 4.3, 4.1 and, 4.2, ί = 3f 4 or 7. By 4.11, ί = 7 impliesG ~ G* and so gives strict inequality. For t = 3, 4, see end of § 2.

392 A. W. MASON AND W. W. STOTHERS

5* Imprimitive factor groups* By 4.4, K* will be imprimitivewhen l/2ί < k < t — 1. For this range, we have one general result.

THEOREM 5.1. If k t/2, then # * - <F, £/(?*>, wiίΛ F 2 = 1.

PROOF. By 1.1, the stabilizer of # * has an orbit T = {XfiΓ*, ,Uk~iχH*} of length A?. By [10, p. 44], there is a paired orbit T ofthe same length k. As k ί/2, T = T'.

By [10, p. 45], there is an element FeϋΓ* with VXH* = # * andFfiΓ* - X£Γ*. Then FX = Ur and X"*F = ί7s, for some r, s. Thus,K* = <F, #*>, and F 2 = X ^ ^ X = l^'+ . As H* n XH*X~ι - {1},F 2 - 1.

COROLLARY 5.2. 1/ Λtf is odd, then k < ί/2.

For T(29 3), there are subgroups with k > ί/2 For example,the subgroups 42(2, m), defined in [6], have 42(2, m)* = 42(2, m), sothat ί(42(2, m)) = 3m, &(42(2, m)) = 2m.

If we restrict k further, the imprimitivity can be described moreprecisely. For convenience, we put h = t — k.

THEOREM 5.3. If kΦt — 1, then k Λ2.

Proof. If & £ — 1, then h > 1, and we may suppose that wehave k > max{fe, (h — I)2}.

Consider the action of U on the cosets of H*. It fixes /f*, andpermutes the ί/ XJEΓ* cyclically. We choose Vlf •••, Vh so that

X * = VH* y ^ ^ r * u . . . u F,£Γ* U XiBΓ* U ί/XίT* U U U'-'XH* ,

w h e r e we may assume t h a t Vt = 1.Clearly, F 2 £ P , •••, F^JEΓ* belong to cycles of length a t most

h - 1. Thus, for i = 2, ••-, fe,

with 0 < s(i) ^ fe - 1. If 2 i, i ^ Λ, then

Since s(ΐ)s(i) ^ {h — I)2 < &, the intersection is nontrivial. It canbe shown similarly that, for 1 i ^ h and 0 j ^ & — 1,

V.H^VI1 n (U'x^iUfX)-1 - {1}.

Let [£Γ*] = {FiίΓ*: i = 1, , fe} and let T7e K*. Suppose that, forsome i and j , VJI* - WVάH*. Then, for any r, VJI^V^ and

REMARKS ON A THEOREM OF L. GREENBERG ON THE MODULAR GROUP 393

have nontrivial intersection. Hence [H*] is a blockand so h\k. We put m = kjh.

Let KQ be the subgroup which fixes blocks setwise. If FeJfiΓ0

fixes two cosets belonging to different blocks, we may suppose thatVH* = H*, so V — U9G* for some q. Since V also fixes a coset ofthe form UrXH*, 1.1 shows that V — 1. Thus, no two elements ofKQ have the same effect on H* and on XH*, so that

(4) \KQ\^h*.

The blocks are [#*] and {Ui+jmXH*: j = 0, , Λ - 1} for i = 0, ,m - 1. Thus, t/wJBr*, ••-, Um*-X)H* fix the blocks. None of thesefixes a coset UrXH*. Taking conjugates, we obtain a similar setfor each block, i.e., fixing one element of the block, but none in anyother block. All of these are distinct and nontrivial, so

(5) | JEo |^ l + (Λ

Combining (4) and (5), we get the result.

From 5.3 and 4.3, we obtain

COROLLARY 5.4. If t > ab + 1, then μ S a'Vt\t - tι/z).

LEMMA 5.5. In the notation of 5.3, if k> max(/ι, (h — I)2), then\KQ\ ^h\

Proof. With m - k/h as in 5.3, A - UmG*, B = XUmχ-ιG* fix[/f*], [X£Γ*] respectively, and each has order h. Suppose that wehave ArBs = A^3, with 0 r, β, i, j < h, then A5-** = B8~j. Then,since A fixes Jϊ* and JB fixes XH* and only the identity fixes both,we must have r — i and j = s. Hence, Ko — {ArBs: 0 r, s < h).

LEMMA 5.6. Ifk = h2>l, then h is prime.

Proof. If A, as in 5.5, does not fix [fiΓ*] elementwise, then ithas a conjugate distinct from A which fixes some T JEΓ* not fixedby A. From the description of Ko in 5.5, this conjugate is a powerof A. Considering the effect on [H*], this must be A\ with (ef h) > 1.As (β, fe) > 1, Aβ does not act as a cycle on [XH*]. This is a con-tradiction since a conjugate of A would have this effect.

Thus, A fixes [H*] elementwise and has the effect of an j -cycleon the other blocks. As k = h2, there are h + 1(>2) blocks. Welabel the blocks so that [H*] is block zero, [XH*] block one, and soon. Then we have

A — coct eh ,

394 A. W. MASON AND W. W. STOTHERS

where c0 is 1 and ct an ft-cycle on block i, i = 1, , ft. Similarly,

B = dςd1 dh ,

where dx is 1 and dό an ft-cycle on block j , j = 0 and j = 2, , ft.Suppose that C is a conjugate of A fixing cosets in block two.

Then, for some r, s, C — ArBs. As C acts as an ft-cycle on blockzero, (do)

s is an ft-cycle, so that (s, ft) = 1. Similarly, (r, ft) = 1. Con-sidering the effect on block two, we must have (c2)

r(d2)s = 1, so that

2 = feΓ(?), with (w(2), ft) = 1. For the other blocks, we have cor-responding integers w(2), •••, w(h). Since no element fixes cosets intwo blocks, the w(i) are distinct modulo h. As there are h — 1 ofthem, h is prime.

LEMMA 5.7. With the notation of 5.5, if Go is the subgroup ofT(a, b) corresponding to Ko, then (6?0)* = Go and k(G0) = k/h.

Proof. By definition, (6?0)* is generated over Go by Us, where sis the least positive integer with X~ιUsXU~s e Go. The correspond-ing element of Ko sends # * to VtH* for some i. Thus, USXH* =XVtH*. A s I F ^ e f l f f l ^ G * fixes [XH*] and [H*]. Thus,[7s 6 6?o, so that (Go)* = Go. From the proof of 5.5, k(G0) = n(G0) = k/h.

LEMMA 5.8. If k = h2, then h\ab and h + 1 is a prime power.

Proof. With Go as in 5.7, |Γ(α, 6): 6?0| = kt/h', and fc(6?0) = k/h, soί(<?0) = fc + l( = l + fc(G0)). As in §4, T(a9b)/G0 is Frobenius and(kfh)\ab9 and Λ + 1 is a prime power.

THEOREM 5.9. If k = h2>l, then h = 2 (w#fe X* = St).

Proof. From the previous results, K*/Ko is Frobenius of orderh(h + 1), K0 = Chx Ch, h is prime and H l a prime power.

If FeϋΓ* centralizes ^ 0 , then A F # * - VAH* = Fίί*, where Ais as in 5.5. Since A fixes cosets in [if*] only, V fixes [//*]. Onconsidering conjugates of A, V fixes each block setwise, and sobelongs to Ko. Hence, there is a monomorphism K*/K^ Aut(K0) =GL(2, h). Thus, GL(2, h) has a subgroup which is Frobenius of orderh(h + 1). Its kernel N is elementary abelian of order h + 1.

If h > 2, then ft + 1 = 2s, and iVnSL(2, h) is elementary-abelianof order at least 2s"1. The only element of order 2 in SL(2, h) is— I, but this is central in GL(2, h). Hence, s = 1, which is impossible.

Thus, ft = 2, and K* the semidirect product of C2 x C2 byGL(2, 2), i.e., Jf* = S4. It is clear that this will occur if and onlyif one of a and b is even and the other divisible by 3, see 4.8 and

REMARKS ON A THEOREM OF L. GREENBERG ON THE MODULAR GROUP 395

4.9. For the modular group, Γ/Γ(4) ~ S4.

It follows that we must have strict inequality in 5.3, at leastwhen t > 6.

6* A final remark* Our results can be restated as results onfinite groups, e.g.,

THEOREM 6 1. If K is a noncyclίc (a, b, k)-group, with (α, 6) = 1,then,

—kΔ

If, in addition, K is simple, then

The second part is trivial when we observe that, in an obviousnotation, K = K* whenever the former is simple.

We should like to express our gratitude to Dr. N. K. Dickson(University of Glasgow) for his assistance with various aspects ofthis paper, particularly with the group theory in § 4.

REFERENCES

1. R. W. D. Accola, On the number of automorphisms of a closed Riemann surface,Trans. Amer. Math. Soc, 131 (1968), 398-408.2. D. Gorenstein, Finite Groups, Harper & Row, New York, (1968).3. L. Greenberg, Note on normal subgroups of the modular group, Proc. Amer. Math.Soc, 17 (1966), 1195-1198.4. A. M. Macbeath, Fuchsian Groups, Queen's College, Dundee, (1961).5. A. W. Mason, On a theorem by Leon Greenberg, Proc. Amer. Math. Soc, 23 (1969),18-23.6. 1 Lattice subgroups of free congruence groups, Glasgow Math. J., 10(1969), 106-115.7. M. Newman, Classification of normal subgroups of the modular group, Trans.Amer. Math. Soc, 126 (1967), 267-277.8. f A note on modular groups, Proc. Amer. Math. Soc, 14 (1963), 124-125.9. D. S. Passman, Permutation Groups, Benjamin, New York, (1968).10. H. Wielandt, Finite Permutation Groups, Academic Press, New York, (1964).

Received April 25, 1977.

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Su-Shing Chen, Weak rigidity of compact negatively curved manifolds . . . . . 273Heinz Otto Cordes and D. A. Williams, An algebra of pseudodifferential

operators with nonsmooth symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Herbert Paul Halpern, Normal expectations and integral decomposition of

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automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333Dean G. Hoffman and David Anthony Klarner, Sets of integers closed under

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application to invariant means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369Jack W. Macki, A singular nonlinear boundary value problem . . . . . . . . . . . . . 375A. W. Mason and Walter Wilson Stothers, Remarks on a theorem of L.

Greenberg on the modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Kevin Mor McCrimmon, Peirce ideals in Jordan algebras . . . . . . . . . . . . . . . . 397John C. Morgan, II, On the absolute Baire property . . . . . . . . . . . . . . . . . . . . . . 415Gerard J. Murphy, Commutative non-Archimedean C∗-algebras . . . . . . . . . . . 433Masafumi Okumura, Submanifolds with L-flat normal connection of the

complex projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447Chull Park and David Lee Skoug, Distribution estimates of barrier-crossing

probabilities of the Yeh-Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455Irving Reiner, Invariants of integral representations . . . . . . . . . . . . . . . . . . . . . . 467Phillip Schultz, The typeset and cotypeset of a rank 2 abelian group . . . . . . . 503John Brendan Sullivan, Representations of Witt groups . . . . . . . . . . . . . . . . . . . 519Chia-Chi Tung, Equidistribution theory in higher dimensions . . . . . . . . . . . . . 525Toshio Uda, Complex bases of certain semiproper holomorphic maps . . . . . . 549

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