A descent principle in modular subgroup arithmetic

Transcription

1 Journal of Pure and Applied Algebra 203 (2005) A descent principle in modular subgroup arithmetic Peter J. Cameron, Thomas W. Müller School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom Received 28 September 2003; received in revised form 16 February 2005 Available online 10 May 2005 Communicated by E.M. Friedlander Abstract We establish and comment on a surprising relationship between the behaviour modulo a prime p of the number s n (G) of index n subgroups in a group G, and that of the corresponding subgroup numbers for a subnormal subgroup of p-power index in G. One of the applications of this result presented here concerns the explicit determination modulo p of s n (G) in the case when G is the fundamental group of a finite graph of finite p-groups. As another application, we extend one of the main results of the second author s paper (ForumMath, in press) concerning the p-patterns of free powers G q of a finite group G with q a p-power to groups of the more general form H G q, where H is any finite p-group Elsevier B.V. All rights reserved. MSC: 20E07; 20E06 1. Introduction For a group G and a positive integer n, denote by s n (G) the number of index n subgroups in G. 1 We call G an FSG-group if s n (G) is finite for all n; for instance, finitely generated groups and groups of finite subgroup rank have this property. Modular subgroup arithmetic, a chapter in the theory of subgroup growth, deals with divisibility properties of the sequence {s n (G)} n 1 or related subgroup counting functions and their connection with the algebraic Corresponding author. addresses: p..cameron@qmul.ac.uk (P.J. Cameron), t.w.muller@qmul.ac.uk (T.W. Müller). 1 The reader should be warned that, in the literature on subgroup growth, s n (G) often denotes the number of subgroups in G of index at most n, that is, the summatory function of s n (G) in our notation /$ - see front matter 2005 Elsevier B.V. All rights reserved. doi: /.paa

2 190 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) structure of the underlying group G; cf. the recent book [10] by Lubotzky and Segal for more background information. In general, divisibility properties of subgroup counting functions appear to be rather peculiar to the particular group under investigation, and (unlike their growth behaviour) tend to be severely distorted when passing to a subgroup of finite index. Example. Consider the cartesian map from the modular group G = PSL 2 (Z) C 2 C 3 onto C 2 C 3 C 6. By a theoremof Nielsen, the kernel of this map is free of rank 2; cf. [11] and [17]. Moreover, by a theoremof Stothers [20], s n (PSL 2 (Z)) 1 mod 2 n = 2 σ+1 3orn = 2(2 σ+1 3) for some σ 1. On the other hand, it follows fromm. Hall s recursion formula [8, Theorem5.2] s n (F r ) = n(n!) r 1 ((n μ)!) r 1 s μ (F r ), (n 2,s 1 (F r ) = 1) that s n (F 2 ) is always odd. 0<μ<n Against this background it is rather surprising that a non-trivial positive result in this direction does in fact exist (see Theorem1 below). Given a prime p and an FSG-group G, define the p-pattern Π (p) (G) of G to be the family of sets Π (p) (G) ={Π (p) 1 (G), Π(p) 2 (G),...,Π(p) p 1 (G)}, where Π (p) (G) := {n N : s n (G) mod p}, 0 <<p, in particular, Π G := Π (2) 1 (G) is the parity pattern of G. The main purpose of this paper is to draw attention to the following remarkable result. Theorem 1 (Descent Principle). Let p be a prime, G an FSG-group, and let H G be a subnormal subgroup of index p r. Then Π (p) (G) = p r Π (p) (H) p ρ (Π (p) (H) (N pn)), 0 <<p. (1) 0 ρ<r Equivalently, if X G,p (z) denotes the mod p proection of the series n 0 s n+1(g)z n, and if X H,p (z) is the corresponding GF(p)-series for the group H, then, under our assumptions, X G,p (z) = r r 1 z pρ 1 X H,p (z pρ ) + ρ=0 ρ=0 z pρ+1 1 X (p 1) H,p (zpρ ). (2) Theorem1 follows quickly fromthe main result of [15], where the conclusions (1) and (2) are established under the extra hypotheses that H is normal in G, and that G/H is cyclic; cf. Section 2 for more details. As the above example demonstrates, the assumption in Theorem 1 that (G : H) be a prime power cannot be weakened.

3 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) In Sections 3 and 4, we present two applications of Theorem1. First, consider the fundamental group G of a finite graph (G( ), Y ) of finite p-groups. If G contains a free subnormal subgroup F of index m G = lcm{ G(v) :v V(Y)}, then s n (G) is periodic modulo p, and its p-pattern is determined completely by that of s n (F); cf. Theorem2. Existence of such a free subnormal subgroup F is not guaranteed, and we provide various sufficient conditions, one of which involves homogeneity; we use the classification of finite homogeneous groups due to Cherlin and Felgner [3]. As another application, we extend one of the main results of [16] concerning the p-patterns of free powers G q of a finite group G with q a p-power to groups of the more general form H G q, where H is any finite p-group; cf. Theorem3. We thank the referee, whose comments have improved both the substance and the presentation of the material in Section Remarks on the proof of Theorem 1 We concentrate on Eq. (1); the equivalence of (1) and (2) was already established in [15] (see the end of Section 2 in that paper). First note that Theorem1 has a straightforward reduction to the case of prime index. Indeed, suppose that for an FSG-group G and a normal subgroup H G of index p (a prime), we know that Let Π (p) (G) = pπ (p) (H) (Π (p) (H) (N pn)), 0 <<p. (3) H = H 0 H 1 H r = G be a normal series for H with (H i : H i 1 ) = p for all 1 i r (such a normal series exists since, by Frobenius generalization of Sylow s third theorem [5, Section 4, Theorem1], every non-trivial finite p-group contains a normal subgroup of index p). Then, by (3), we have Π (p) (H i ) = pπ (p) (H i 1 ) (Π (p) (H i 1 ) (N pn)) (1 i r, 0 <<p), (4) and, using (4), an immediate induction on i shows that, for 1 i r and 0 <<p, Π (p) (H i ) = p i Π (p) (H) p ρ (Π (p) (H) (N pn)), 0 ρ<i whence (1). Validity of Eq. (1) in the case where H G and (G : H)=p follows immediately from [15, Theorem1]. Here, we briefly sketch an alternative proof of (3) generalizing an argument in [14]. As in the proof of [14, Proposition 4] one observes that a subgroup G G is of index n in G and not contained in H if, and only if, G S( H), where (H: H)=n S( H) := { G G : G H = H and GH = G}.

4 192 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) It follows that s n (G) = (H: H)=n { S( H) + sn/p ( H), p n, 0, p n. (5) Fix an element ζ with G= H, ζ. Given a subgroup H of index n in H and a right transversal 1 = h 0, h 1,...,h n 1 for H in H, then the elements g μ,ν := h μ ζ ν, (0 μ <n,0 ν <p), forma right transversal for H in G. A subgroup G S( H) must be of the form G = G μ = Hg 0,0 with some vector 0<<p Hg μ, μ = (μ 1, μ 2,...,μ p 1 ) {0, 1,...,n 1} p 1 and such a set G μ G is a member of S( H) if, and only if, G μ is a subgroup of G. The necessary and sufficient condition for the last assertion to hold is that g Hg μ, μk,k = Hg μ+k,+k, 0,k <p with μ 0 := 0 and reduction (indicated by an overstroke) being modulo p. It follows fromthis analysis that the cardinality of the set S( H) equals the number of subgroups in N G ( H)/ H of order p, which are not contained in N H ( H)/ H. Applying Frobenius theorem 2 concerning the number of solutions of the equation X m = 1 in a finite group G in the case when m = p and G = N G ( H)/ H or G = N H ( H)/ H, we deduce that S( H) { 1, p (NG ( H) : H) and p (N H ( H) : H), 0, otherwise, mod p, which, in conunction with (5), yields { sn/p (H), p n, s n (G) Ω n + 0, p n, mod p, (6) where Ω n := { H H : (H : H) = n, p (N G ( H) : H), p (N H ( H) : H)}. 2 cf. [5, pp ] and [6].

5 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) Denote by U n (H) the set of subgroups of index n in H. Then, making use of the action by conugation of H on Ω n and that of G on U n (H)\Ω n, we find that { sn (H), p n, Ω n mod p, 0, p n, which, when combined with (6), gives s n (G) s n/(n,p) (H), mod p, whence (3). 3. Divisibility properties determined by free normal subgroups The category of graphs used in this section is described in Serre s book [19]. Let (G( ), Y ) be a finite graph of finite groups with fundamental group G = π 1 (G( ), Y ), and let p be a prime. Moreover, denote by V(Y)and E(Y) the set of vertices respectively (geometric) edges of Y, and let m G be the least common multiple of the orders of the finite subgroups in G, that is, m G = lcm { G(v) :v V(Y) }. The free rank μ(g) of G is defined as the rank of a free subgroup in G of index m G (such subgroups always exist; cf., for instance, [19, Lemmas 8 and 10]). It is connected with the rational Euler characteristic χ(g) of G via μ(g) + m G χ(g) = 1 (7) and the latter quantity can be computed in terms of the graph of groups (G( ), Y ) by means of the formula χ(g) = 1 G(v) 1 G(e), (8) v V(Y) e E(Y) cf. [1, Chapter IX, Proposition 7.3] or [18, Proposition 14]. If G has a free subnormal subgroup F of index m G a p-power, then every vertex group G(v) must be of p-power order; and if χ(g) 0, then any free subnormal subgroup F of index m G has rank rk(f) = μ(g) 1, and, by Theorem1, the p-pattern of G is determined via (1) by the p-pattern of F. Consequently, all conclusions of [15, Theorem2] remain valid in this more general situation, and we obtain the following. Theorem 2. Let p be a prime, (G( ), Y ) a finite graph of groups all of whose vertex groups are of p-power order, and let G be its fundamental group. Let m G = p r, and suppose that G contains a subnormal free subgroup of index m G, and that χ(g) 0. Then (i) the function s n (G) is periodic modulo p, (ii) for p = 2 we have Π G = N,

6 194 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) (iii) for p = 3 and μ(g) odd we have Π (3) 1 (G) = N, (iv) for p=3 and μ(g) even, s n (G) is periodic modulo 3 with period 8 3 r. More precisely, in this case s n (G) 1 mod 3 if and only if n is congruent mod 8 3 r to one of the 3 r+1 numbers 0, 3 r 1, 3 r, 8 3 r 1, 3 r+1, 11 3 r 1, 16 3 r 1, 17 3 r 1, 19 3 r 1, 3 ρ (1 + 24λ), 8 3 ρ (1+3λ), 3 ρ (11+24λ), 8 3 ρ (2+3λ), 3 ρ (17+24λ), 3 ρ (19+24λ) with 0 ρ <r 1 and 0 λ < 3 r ρ 1 ; and s n (G) 2 mod 3 if and only if n is congruent mod 8 3 r to one of the 3 r+1 numbers 4 3 r 1, 5 3 r 1, 7 3 r 1, 4 3 r, 13 3 r 1, 5 3 r, 20 3 r 1, 7 3 r, 23 3 r 1, 4 3 ρ (1+6λ), 4 3 ρ (5 + 6λ), 3 ρ (5 + 24λ), 3 ρ (7 + 24λ), 3 ρ ( λ), 3 ρ ( λ) with 0 ρ <r 1 and 0 λ < 3 r ρ 1. The usefulness of Theorem2 depends on our being able to verify the hypothesis that G contains a subnormal free subgroup of index m G. The remainder of this section is devoted to this last problem. Rather than attempt to state a very general result here, we isolate the essential part of the argument in the next two lemmas, followed by several applications. Lemma 1. Let (G( ), Y ) be a finite tree of finite groups, and set G = π 1 (G( ), Y ) and m = m G. For e E(Y), and a vertex v in the boundary of e, let α (e,v) be the embedding of G(e) into G(v) given by the tree of groups (G( ), Y ). Assume that there is a vertex v 0 V(Y)with the property that G(v 0 ) =m. Set G = G(v 0 ), and denote by Ψ = Ψ(G( ), Y ) the set of all homomorphisms ψ : G G such that the restriction of ψ to any vertex-group is inective. Then the following hold: (i) Ψ is non-empty if and only if there is a family {ψ v : G(v) G} v V(Y) of inective homomorphisms such that, for every e E(Y), we have ψ v α(e,v)g(e) = ψ v α(e,v )G(e), (9) where v and v are the two vertices bounding e. (ii) The number of free normal subgroups having index m in G is Ψ / Aut(G). Proof. (i) For v V(Y), define ψ v to be ψ G(v), for all v V(Y). Then Eq. (9) follows fromthe definition of a tree of groups. Conversely, if the homomorphisms ψ v exist and satisfy Eq. (9), then there is a homomorphism ψ : G G whose restriction to G(v) is ψ v ; by definition, ψ Ψ. (ii) Aut(G) acts naturally on Ψ via ψ α := α ψ (ψ Ψ, α Aut(G)) and, since each ψ Ψ is surective, this action is free; thus Ψ/Aut(G) = Ψ / Aut(G). Now let F be a free normal subgroup of G of index m. Then G(v 0 ) F=1 and G(v 0 )F=G, so G/F G, and the canonical proection map π from G to G belongs to Ψ. Sending F

7 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) to [π], the orbit of π under Aut(G), gives a well-defined map φ :{F G : F free, (G : F) = m} Ψ/Aut(G). Moreover, the kernel of any member ψ of Ψ is a free normal subgroup of G of index m (see, for instance, [4, Chapter II, Theorem1.3]), and the proection map π : G G/ ker(ψ) G differs from ψ only by an automorphism of G, so[π]=[ψ], and φ is surective. Finally, if two free normal subgroups F 1, F 2 of G of index m have proections π : G G/F i G only differing by an automorphism of G, then F 1 = F 2,soφ is a biection. Definition. For a group G and a subgroup H G, we define the centralizer C Aut(G) (H) of H in Aut(G) to be C Aut(G) (H) := {α Aut(G) : α(h) = h for all h H}. Lemma 2. Let the hypotheses and notation be as in Lemma 1. Suppose that E(Y) is nonempty and that the set Ψ in Lemma 1 is also non-empty. Choose v to be a terminal vertex of Y different from v 0. Let Y 0 be the subtree of Y induced on V(Y) {v }, and let Ψ 0 = Ψ(G( ) Y0,Y 0 ). Let e be the edge containing v, and let v V(Y 0 ) be the other vertex bounding e. For ψ 0 Ψ 0, let S(ψ 0 ) be the set of inective homomorphisms ψ v : G(v ) G for which ψ α(e,v )G(e ) = ψ 0 α(e,v )G(e ).(This set may be empty). Then Ψ = ψ 0 Ψ 0 S(ψ 0 ). Proof. Let G 0 = π 1 (G( ) Y0,Y 0 ).WehaveG = G 0 G(e )G(v ), the amalgamation being with respect to the canonical embeddings of G(e ) in G 0 and G(v ) respectively. The number of elements of Ψ which restrict to ψ 0 is equal to S(ψ 0 ). The results of these two lemmas are most easily applied when the vertex groups are homogeneous, a concept we discuss next. A group G is termed homogeneous if every isomorphism between finitely generated subgroups is induced by an automorphism of G. This concept arose (for arbitrary first order structures) in model theory in connection with quantifier elimination. For instance, it is known that a finite group is homogeneous if and only if its first order theory has quantifier elimination; cf. [9, Corollary 8.4.2]. The finite homogeneous groups have been classified by Cherlin and Felgner; cf. [2] and [3]. In particular, a finite p-group G is homogeneous if and only if one of the following holds: (i) G C p r C p r for some r, s N 0 = N {0}. }{{} s copies (ii) G Q, the quaternion group of order 8. (iii) G Q. Here, Q is a certain group of order 64, class 2, and exponent 4, which arises for instance as the Sylow 2-subgroup of PSU 3 (4 2 ); cf. [2, Section 2] for more details.

8 196 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) Proposition 1. Let p be a prime. Let (G( ), Y ) be a finite tree of groups such that (i) Every vertex group G(v) is a finite homogeneous p-group. (ii) There is a vertex v 0 V(Y)such that every vertex group is isomorphic to a subgroup of G = G(v 0 ). Set G π 1 (G( ), Y ). Then the number of free normal subgroups of index m G = G in G is / C Aut(G) (G e ) C Aut(G) (G v ), e E(Y) v V(Y) where G v and G e are subgroups of Gisomorphic to G(v) and G(e) respectively, for v V(Y)and e E(Y). Proof. The proof is by induction on E(Y). (The induction hypothesis, in conunction with Lemma 1, asserts that Ψ is equal to Aut(G) times the quantity in the Proposition.) If E(Y) =, then V(Y)={v 0 }, and the formula gives 1/ C Aut(G) (G) =1, which is correct; so Ψ = Aut(G) in this case. Suppose that E(Y) =. Choose v to be a terminal vertex of Y not equal to v 0, let e be the edge containing v and let v be the other vertex in the boundary of e. Since α(e,v )G(e ) α(e,v )G(e ), the sets S(ψ 0 ) appearing in Lemma 2 are all non-empty. Let G be a finite homogeneous group. If H G, then the number of extensions of a given embedding H G to an automorphism of G is C Aut(G) (H ). Hence, if K H G, then the number of extensions of an embedding K G to an embedding H G is C Aut(G) (K) / C Aut(G) (H ). Hence, in Lemma 2, we have S(ψ 0 ) = C Aut(G) (G e ) / C Aut(G) (G v ), independent of ψ 0, and so Ψ = Ψ 0 C Aut(G) (G e ) / C Aut(G) (G v ). On the other hand, the induction hypothesis asserts that / Ψ 0 = Aut(G) C Aut(G) (G e ) e E(Y 0 ) v V(Y 0 ) C Aut(G) (G v ) and combining the last two equations, and applying Lemma 1 again, gives the desired result, since V(Y)= V(Y 0 ) {v } and E(Y) = E(Y 0 ) {e }. Corollary 1. Let p be a prime, (G( ), Y ) a finite tree of groups all of whose vertex groups are of p-power order, G π 1 (G( ), Y ), and let m G = p r. (i) If all vertex groups G(v) are cyclic, then G contains precisely v V(y) φ( G(v) ) / e E(y) φ( G(e) ) φ(m G ) free normal subgroups of index m G, where φ is Euler s totient function.

9 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) (ii) If all vertex groups G(v) are elementary abelian, then G contains exactly e E(y) [ G(e) r d e GL r de (p) ] e V(y) [ G(v) r d v GL r dv (p) ] free normal subgroups of index m G, where d σ = dim p G(σ), σ V(Y) E(Y). Proof. The hypotheses of Proposition 1 are satisfied in both cases. Note that, if G is a finite homogeneous group and H a subgroup of G, then we have N Aut G (H )/C Aut G (H ) Aut(H ), and hence where C Aut(G) (H ) = Aut(G) #(G, H ) Aut(H ), N Aut(G) (H ) := {α Aut(G) : α(h ) = H } and with #(G, H ) = Aut(G) / N Aut(G) (H ) the number of subgroups of G isomorphic to H. For (i), we have Aut(G) =φ( G ) and #(G, H ) = 1 whenever G is a cyclic group and H G. Proposition 1 shows that the number of free subgroups of G of index m G is e E(y) φ(m G)/φ( G(e) ) v V(y) φ(m G)/φ( G(v) ), which is equal to the value claimed, since V(Y) = E(Y) +1. (Note that this result is also proved in [15, Lemma 1].) For (ii), if G is elementary abelian of order p r, and H is a subgroup of order p s, then we have Aut(G) = GL r (p) =(p r 1)(p r p) (p r p r 1 ) and #(G, H ) = [ ] r = (pr 1)(p r p) (p r p s 1 ) s p (p s 1)(p s p) (p s p s 1 ). Hence, if G =p r, H G, and H =p s, then C Aut(G) (H ) = GL r s (p) p s(r s) = GL r s (p) H r s. Hence, the result follows fromproposition 1. Corollary 2. (i) If, in Proposition 1, all the vertex groups are isomorphic to G = Q then G contains precisely 24 a(g) 4 b(g) free normal subgroups of index m G = 8, where a(g) := {e E(Y) : G(e) < 4} and b(g) := {e E(Y) : G(e) =4}.

10 198 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) (ii) If, in Proposition 1, all the vertex groups are isomorphic to G = Q, the number of free normal subgroups in G of index m G = 64 equals where a(g) 5120 b(g) 2560 c(g) 256 d(g) 128 e(g) 32 f(g) 16 g(g) 4 h(g), a(g) := {e E(Y) : G(e) = 1}, b(g) := {e E(Y) : G(e) =2}, c(g) := {e E(Y) : G(e) C 2 C 2 }, d(g) := {e E(Y) : G(e) C 4 }, e(g) := {e E(Y) : G(e) =8}, f(g) := {e E(Y) : G(e) C 4 C 4 }, g(g) := {e E(Y) : G(e) H 16 }, h(g) := {e E(Y) : G(e) =32}. Here, H 16 = a,b a 4 = b 4 = 1,a b = a 1. Proof. Since all vertex groups are isomorphic to G, wehavec Aut(G) (G v ) = 1 for all v V(Y). The assertions of the corollary follow immediately from Proposition 1, once the orders of the corresponding centralizers C Aut(G) (G e ) are known. For the second part, these have been found with the help of the computer algebra system GAP [7]. Remark. A finite group G is homogeneous if and only if, for every finite tree of groups (G( ), Y ) with all vertex stabilizers isomorphic to G, G = π 1 (G( ), y) contains a free normal subgroup of index m G. Indeed, the forward implication follows immediately from Lemma 1. If, on the other hand, G is not homogeneous, then there exists an isomorphism between two subgroups H 1,H 2 of G, which is not induced by an automorphism. We can then formthe amalgamg=g H G, where the abstract group H is identified with H 1 in the left factor and with H 2 in the right factor, in such a way that the isomorphism between H 1 and H 2 induced by these embeddings is the given isomorphism. Then G does not possess a free normal subgroup of index m G = G. For,ifF were such a subgroup, then G/F G, and the two embeddings of H in G would coincide in this quotient, which (by assumption) is not the case. We now describe a simple example to illustrate that, even if the vertex groups are not homogeneous, the counting may still be possible. Let (G( ), Y ) be a tree of groups all of whose vertex groups are isomorphic to D 4, the dihedral group of order 8, and let G be its fundamental group. Call an edge e E(Y) wild, if G(e) contains an involution which is identified with the central involution in one of its corresponding vertex groups, and a non-central involution in the other. If e is not wild, we call it inner if G(e) is embedded in the cyclic subgroup of order 4 of G(v) for v on e, and outer otherwise. Proposition 2. Let G be as above. Then G has a free normal subgroup of index m G = 8 if and only if the tree of groups (G( ), Y ) does not contain a wild edge. In the latter case,

11 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) the number of free normal subgroups of index m G equals 2 o(g) 8 φ( G(e) ), where e E(Y) G(e) inner o(g) := {e E(Y) : G(e) outer}. Proof. Necessity of the stated existence criterion is clear. In the positive direction, the condition that no edge is wild guarantees that, in the notation of Lemma 2, the sets S(ψ 0 ) are all non-empty, and have cardinality 2 if e is outer and 8/φ( G(e) ) if e is inner. Define the type τ(g) of a finitely generated virtually free group G π 1 (G( ), Y ) as the tuple τ(g) = (m G ; ζ 1 (G),...,ζ κ (G),...,ζ mg (G)), where the ζ κ (G) are integers indexed by the divisors of m G, given by ζ κ (G) = {e E(Y) : G(e) κ} {v V(Y): G(v) κ} with V(Y) and E(Y) as above. We have ζ κ (G) 0 for κ <m G and ζ mg (G) 1 with equality occurring in the latter inequality if and only if Y is a tree; cf. [12, Lemma 2] and [13, Proposition 1]. It can be shown that the type τ(g) is in fact an invariant of the group G, that is, independent of the particular decomposition of G in terms of a graph of groups (G( ), Y ), and that two virtually free groups G 1 and G 2 contain the same number of free subgroups of index n for each positive integer n if and only if τ(g 1 ) = τ(g 2 ); cf. [12, Theorem2]. It follows from(8) that the Euler characteristic of G can be expressed in terms of the type via χ(g) = m 1 G φ(m G /κ)ζ κ (G). (10) κ m G Eqs. (7) and (10) imply in particular that, if two virtually free groups have the same number of free index n subgroups for each n, then their Euler characteristics respectively free ranks must coincide. For a finitely generated virtually free group G and a prime p define the p-rank μ p (G) of G by means of the formula μ p (G) = 1 + φ(m G /κ)ζ κ (G). p κ m G Moreover, denote by f λ (G) the number of free subgroups in G of index λm G. Proposition 3. Let p be a prime, (G( ), Y ) a finite graph of groups all of whose vertex groups are non-trivial finite p-groups, and let G = π 1 (G( ), Y ). Then the following assertions are equivalent: (i) f 1 (G) / 0modp,

12 200 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) (ii) μ p (G) = 0, (iii) G is a free product of the form G H C p C p with s 0 andagrouphof }{{} order m G. s copies Corollary 3. Let p be a prime, and let G = H Cp s be a free product of s 0 copies of the cyclic group of order p and a finite p-group H. Then G contains a normal free subgroup of index m G. Proof. This follows fromthe action by conugation of G on the set of free subgroups of index m G, together with the implication (iii) (i) of Proposition 3. It also follows immediately from Lemma 1. Proof of Proposition 3. The equivalence of (i) and (ii) follows froma discussion of the formula 3 f 1 (G) = m G k ζκ(g), κ m G 1 k m G (m G,k)=κ making use of facts concerning τ(g) mentioned above. Suppose now that μ p (G) = 0. Then Y is a tree, and, after contracting edges of Y corresponding to trivial amalgamations if necessary, we may assume that (G( ), Y ) is normalized, that is, G(e) = G(v) for all e E(Y) and v e. For a positive integer n, denote by e n,v n the number of edges e E(Y) respectively vertices v V(Y)whose associated group G(e) respectively G(v) has order n, define an arithmetic function f (n) via f (n) = (e ν v ν ), n 1 ν n and let m G = p r. Then, for 0 ρ r, f(p ρ ) = { e1, ρ = 0, 1, ρ = r, 0, otherwise and, by Möbius inversion, (11) e n v n = ν n μ(ν)f (n/ν), n 1, (12) where μ is the classical Möbius function. Since our claim(iii) holds for r 1, we may assume that r 2. In the latter case, we find from(11) and (12) that { e1, ρ = 1, e p ρ v p ρ = 0, 1 < ρ <r, (13) 1, ρ = r. 3 cf. formulae (3) and (11) in [12].

13 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) Using the facts that (G( ), Y ) is normalized and that Y is a tree (hence, in particular, does not contain loops), we find from(13) that e p r = 0, therefore v p r = 1, e p r 1 = 0, therefore v p r 1 = 0,. e p 2 = 0, therefore v p 2 = 0, e p = 0, therefore v p = e 1. It follows that all edge groups are trivial, that is, G is the free product of its vertex groups, and that V(Y) contains precisely one vertex v 0 with G(v 0 ) =p r and e 1 0 vertices v satisfying G(v) C p, whence (iii). Since the implication (iii) (ii) is trivial, the proof of Proposition 3 is complete. 4. The groups G(G,H,q) For a finite group G, a prime p, and p-powers q, q with q q>1, let G = G(G,H,q)= H G } G {{}, (14) q copies where H is of order q. Put G := G(G, 1,q) G q. It follows fromthe normal formtheorem applied to the free product H G that G(G,H,q)is a split extension of the group H = G h : h H G q q = G(G, 1,q q) by H; in particular, the groups G and H satisfy the hypotheses of Theorem1, and (1) yields the reduction formula Π (p) (G(G,H,q))= qπ (p) (G(G, 1,q q)) σ σ q σ< q (Π (p) (G(G, 1,q q)) (N pn)), 0 <<p. (15) Formula (15) allows us to translate results concerning the groups G(G, 1, q) obtained in [16] into results for groups of the more general form (14). Since, for the most part, this translation process is entirely straightforward, and whatever extra arguments are needed can be found in [16, Section 8], we shall leave this task to the reader. As an example, we state the generalization of [16, Theorem12], which provides a remarkably explicit combinatorial description of the p-pattern Π (p) (G(G,H,q)) under a certain assumption on G. Theorem 3. Let Gbe a finite group, p a prime, let q and q be p-powers such that q q>1, and let H be a group of order q. Assume that s d (G) 0(p) for all d N with d / 1(p)

14 202 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) (that is, G Fin(p) in the notation of [16]). Then we have Π (p) (G(G,H,q))= σθ () G,q, q, σ q 0 <<p, where Θ () G,q, q consists of all positive integers n 1modpq q such that the sum n N τ 0 d G,p n= n 1 pq q is congruent to modulo p. ( ) r 1 + (q q 1)(n 1)/(q q) (s n, 1 + (q q 1)(n 1)/(q q) n di (G)) n i Here, the vector d G,p N r attached to the group G and prime p is defined as ( d1 1 d G,p := p, d ) 2 2 p,...,d r 1, p where 1 = d 0 <d 1 < <d r = G is the collection in increasing order of those positive integers d for which s d (G) / 0(p). Also, if n = (n 1,...,n r ) is a vector of positive integers with sum n, and N n, then ( N n,n n denotes the multinomial coefficient N! n 1!n 2! n r!(n n )!. ) i=1 References [1] K.S. Brown, Cohomology of Groups, Springer, New York, [2] G.L. Cherlin, U. Felgner, Homogeneous solvable groups, J. London Math. Soc. 44 (2) (1991) [3] G.L. Cherlin, U. Felgner, Homogeneous finite groups, J. London Math. Soc. 62 (2) (2000) [4] W. Dicks, Groups, Trees, and Proective Modules, Lecture Notes in Mathematics, vol. 790, Springer, Berlin, [5] G. Frobenius, Verallgemeinerung des Sylow schen Satzes, Berliner Sitzungsberichte (1895) [6] G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie, Berliner Sitzungsberichte (1903) [7] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.3; Aachen, St. Andrews, 2002, gap [8] M. Hall, Subgroups of finite index in free groups, Canad. J. Math. 1 (1949) [9] W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, [10] A. Lubotzky, D. Segal, Subgroup Growth, Progress in Mathematics, Birkhäuser, Basel, [11] R. Lyndon, Two notes on Rankin s book on the modular group, J. Austral. Math. Soc. 16 (1973) [12] T. Müller, Combinatorial aspects of finitely generated virtually free groups, J. London Math. Soc. 44 (2) (1991) [13] T. Müller, A group-theoretical generalization of Pascal s triangle, European J. Combin. 12 (1991)

15 P.J. Cameron, T.W. Müller / Journal of Pure and Applied Algebra 203 (2005) [14] T. Müller, Parity patterns in Hecke groups and Fermat primes, in: T.W. Müller (Ed.), Groups: Topological, Combinatorial and Arithmetic Aspects, LMS Lecture Note Series 311, Cambridge University Press, Cambridge, 2004, pp [15] T. Müller, Modular subgroup arithmetic and a theorem of Philip Hall, Bull. London Math. Soc. 34 (2002) [16] T. Müller, Modular subgroup arithmetic in free products, Forum Math., in press. [17] J. Nielsen, The commutator group of the free product of cyclic groups, Mat. Tidsskr. B (1948) [18] J.-P. Serre, Cohomologie des groupes discrets, Ann. Math. Studies, vol. 70, Princeton University Press, Princeton, [19] J.-P. Serre, Trees, Springer, Berlin, [20] W. Stothers, The number of subgroups of given index in the modular group, Proc. Roy. Soc. Edinburgh 78A (1977)