The p-coclass of a group

D.F. Holt Mathematics Institute University of Warwick Coventry CV4 7AL

England

W. Plesken Lehrstuhl B für Mathematik RWTH Aachen Templergraben 64 5100 Aachen

November 18, 1997

Proposed running head:

The p-coclass of a group.

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D.F. Holt Mathematics Institute University of Warwick Coventry CV4 7AL England

Let p be a prime and P a finite p-group. The class cl(P) of P is defined to be the length of its lower central series, and the coclass cocl(P) is defined by cocl(P) = n−cl(P), where |P | = p^{n}. In a series of papers (see, for example, [LeN 80] and [Don 87]) Leedham-Green and others showed that, while it is virtually impossible to produce a reasonable description of p-groups of low class (even as low as 2), a highly satisfactory theory of pgroups of small coclass can be developed. It turns out that, in some strong sense, ’most’ finite p-groups of a fixed coclass are factor groups of a finite number of infinite soluble pro-p-groups, at least modulo a ’small’ normal subgroup. (See, for example [DSMS 91] for background information on pro-p-groups.)

During the course of their investigations into perfect groups [HoP 89], the authors observed that, statistically, the finite perfect groups appear to be dominated by extensions of p-groups by relatively small finite simple and quasisimple groups. Indeed, it is conjectured in [Hol 89] that, in some appropriate sense, ’most’ finite perfect groups are extensions of 2-groups of class 2 by A_{5}. It therefore seemed natural to ask whether we could generalize the notion of coclass to extensions of p-groups by a small fixed group and whether this would lead to a theory similar to that developed by Leedham-Green and others. The purpose of this article is to suggest a definition, to prove one or two easy results, and to provide some examples that indicate the difficulties inherent in this task and reveal some possible lines of attack.

Definition 1 (i) Let X be a finite group, p a prime, K = O_{p}(X) and G = X/K. Then we define the p-coclass of X, cocl_{p}(X) = f − c, where f is the number of G-composition factors of K and c = cl(K).

(ii) Let p be a prime and X a virtual pro-p-group (i.e. it has a normal pro-p-subgroup of finite index). Let K be the maximal normal pro-p-subgroup of X and G = X/K (so G is finite and O_{p}(G) = 1). Then the p-coclass of X is defined as

cocl_{p}(X) = lim_{i→∞} cocl_{p}(X/γ_{i}(K))

if this exists, and infinity otherwise, where γ_{i}(K) are the terms of the lower central series of K as pro-p-group (i.e. γ_{1}(K) = K and γ_{i}(K) is the closure of [K, γ_{i−1}(K)] for i > 1).

Notice that in both cases cocl_{p}(X) = ∑_{i}(a_{i} − 1) where a_{i} is the number of Gcomposition factors of γ_{i}(K)/γ_{i+1}(K). Thus, in the profinite case, cocl_{p}(X) is finite if and only if each a_{i} is finite and a_{i} = 1 for all sufficiently large i. In this case, it follows at once that K has finite width, in the sense that |γ_{i}(K)/γ_{i+1}(K)| is bounded. If X is itself a p-group (or pro-p-group), then G = 1, and our definition agrees with the usual definition of coclass. However, when G = 1, we always have cocl_{p}(X) > 0 (at least when |X| > p), whereas when |G| > 1 we can have coclass 0 in nontrivial examples. This suggests the following problem. Choose G and p, and attempt to classify finite and profinite groups of small coclass k. Unfortunately, this does not seem easy, except in very small examples. Thus, if we choose G = A_{5}, p = 2 and coclass k = 0. Then we find, by looking at the tables in Chapter 5 of [HoP 89], that G must be one of the small number of finite groups in the diagram below. The names of the groups are as in [HoP 89]; in particular, 2^{1}, 2^{4} and 2^{4′} represent the three simple modules for A_{5} over IF_{2}, where 2^{4} is the matrix representation over IF_{4} and 2^{4′} comes from the natural permutation representation.

[figure: A_{5}2^{4}C2^{4′}; A_{5}2^{4} (split); A_{5}; A_{5}2^{1} = SL(2, 5); A_{5}2^{4′} (split); A_{5}2^{4′}C2^{1}; A_{5}2^{4′}CN2^{1}; 2^{4′}; 2^{4}; 2^{1}; 2^{4′}; 2^{1}; 2^{1}]

However, if we choose k = 1, then it is easy to construct profinite examples. Let H = SL(2, 5). Let L be either of the two four-dimensional lattices for H over the 2-adic integers ZZ_{2}, and let X be an extension of L by H. (There are three possibilities for X; a split extension for either lattice, and a nonsplit extension for the non-absolutely irreducible lattice.) Note that K is an extension of L by the centre < t > of H; since t acts by inversion on L, it is clear that cocl_{2}(X) = 1, and γ_{i}(K) = 2^{i−1}L for all i > 1. We shall see shortly that these examples are typical for the case in which K is a soluble pro-p-group. First, we establish an easy but important lemma.

Lemma 1 Let X be a virtual pro-p-group and p a prime such that cocl_{p}(X) = k is finite. Let i = i_{0} be minimal such that cocl_{p}(X/γ_{i}(K)) = k. Then for any nontrivial closed normal subgroup N of X with N ⊆ γ_{i_{0}}(K), there is an i such that N = γ_{i}(K).

Proof: Since the γ_{i}(K) form a basis for the open neighbourhoods of 1 in K ([DSMS 91], Proposition 1.16) we can find i such that N ⊆ γ_{i}(K) but N ⊈ γ_{i+1}(K). Since i ≥ i_{0}, γ_{i}(K)/γ_{i+1}(K) is irreducible as a G-module, and so γ_{i}(K) = Nγ_{i+1}(K) and similarly, since γ_{j−1}(K)/γ_{j}(K) is an irreducible G-module for all j > i, we get γ_{i}(K) = Nγ_{j}(K). But now, since N is closed, we must have N = γ_{i}(K). q.e.d.

We can now establish the following result in the case when K is soluble.

Theorem 1 Let X be a virtual pro-p-group in which the maximal normal pro-p-subgroup K is soluble (as a pro-p-group), and suppose that cocl_{p}(X) is finite. Then X has a maximal finite normal subgroup N, and X/N is isomorphic to a p-adic space group.

Proof: Since K is a soluble pro-p-group, it has a nontrivial closed normal abelian subgroup, and it follows from the above lemma that γ_{i}(K) is abelian for some i > i_{0}. Since γ_{i}(K) has finite index in γ_{1}(K), which is finitely generated (as a pro-p-group), γ_{i}(K) must be finitely generated ([DSMS 91], Proposition 1.7). Furthermore, the subgroup W =< x ∈ γ_{i}(K) | x^{p} = 1 > of γ_{i}(K) is the kernel of a continuous map, and hence closed, and it follows from Lemma 1 that N = 1, for otherwise N = γ_{j}(K) would be finite for some j. Thus γ_{i}(K) is torsion-free, and so γ_{i}(K) ≅ ZZ_{p}^{n} for some n. The result now follows from Proposition 2.3.12 of [HoP 89]. q.e.d.

Thus, ignoring the finite normal subgroup, a typical example X consists of an extension of a ZZ_{p}H-lattice L by a finite group H acting faithfully on L, where G = H/O_{p}(H) and K is the extension of L by O_{p}(H); of course O_{p}(H) ≠ 1.

Lemma 2 Let L, H, etc. be as above, where X has finite coclass. Then H acts irreducibly and even p-uniserially on L; more precisely, all ZZ_{p}H-sublattices of L are of the form (rad(ZZ_{p}O_{p}(H)))^{i}L = [L,_{i} O_{p}(H)] (where [L,_{1} , O_{p}(H)] = [L, O_{p}(H)] and [L,_{i} O_{p}(H)] = [L, [L,_{i−1} , O_{p}(H)]] for i > 1).

Proof: Choose k ≥ i_{0} such that γ_{k}(K) ⊆ L and α such that p^{α}L ⊆ γ_{k}(K). Then, by Lemma 1, we may assume that p^{α}L = γ_{k}(K). Furthermore, by Lemma 1, all sublattices of p^{α}L are of the form γ_{k+i}(K) = [p^{α}L,_{i} O_{p}(H)] = (rad(ZZ_{p}O_{p}(H)))^{i}p^{α}L, but then, dividing by p^{α}, this second equality must also hold in L. It follows that H acts uniserially and hence irreducibly on L. q.e.d.

If O_{p}(H) is abelian, then we can get a lot more information about the structure of L.

Proposition 1 Let L, H, etc. be as above, where X has finite coclass and suppose that O_{p}(H) is abelian. Let r = rad(ZZ_{p}O_{p}(H)), let n be the dimension of L and let s be the exponent of O_{p}(H). Then for all i ≥ 0, r^{i}L is completely reducible as a ZZ_{p}O_{p}(H)-lattice and each factor r^{i}L/r^{i+1}L has order p^{a} with a = n/(p − 1)p^{s−1}.

Proof: Let L′ = ⨁_{i=1}^{m}(L ∩ e_{i}L), where the e_{i} are primitive idempotents of IQ_{p}O_{p}(H). Then, since the operation of ZZ_{p}O_{p}(H) factorizes over the principal ideal domain ZZ[ζ], with ζ a primitive p^{s}-th root of 1, each L ∩ e_{i}L is completely reducible. Thus L′ and hence each r^{i}L′ is completely reducible. Furthermore, since O_{p}(H) acts faithfully on L, at least one irreducible component has dimension equal to that of ZZ[ζ]; namely (p − 1)p^{s−1}. But the L ∩ e_{i}L are transitively permuted by H and so L′ is H-invariant, and all irreducible components M have the same dimension (p − 1)p^{s−1}. Since |r^{i}M/r^{i+1}M| = p for all i and all such M, each |r^{i}L′/r^{i+1}L′| has the same order p^{a}, where a is the total number of irreducible components. But, by Lemma 2, it is clear that, for some i and α, we have r^{i}L′ = p^{α}L, and so p^{α}L, and hence also L, is completely reducible, and then n = a(p − 1)p^{s−1} as required. q.e.d.

Let us assume now that cocl_{p}(X) = 0. We conjecture that O_{p}(H) cannot be abelian in this case. Suppose that it were. Then, since O_{p}(H) is a factor group of γ_{1}(K)/γ_{2}(K), which consists of a single G-composition factor, O_{p}(H) must be elementary abelian (and so s = 1 in the proposition) and L = γ_{2}(K). Thus, as a G-module, L/rL is a quotient of the wedge-product O_{p}(H)∧O_{p}(H). Let |O_{p}(H)| = p^{l}. Since O_{p}(H) acts faithfully on IQ_{p}L the action must have at least l homogeneous components, and since |O_{p}(H)∧O_{p}(H)| = p^{l(l−1)/2} this immediately implies l ≥ 3, and in fact, by a routine case-by-case analysis, we have shown that l ≥ 4. Finally, we must have O_{p′}(H) = 1, for otherwise (by the Frattini argument), the normalizer of O_{p′}(H) in X would be a complement of K in X, contradicting L = γ_{2}(K).

In view of the above considerations, the next simplest situation where we might look for examples of coclass 0 is with O_{p}(H) an extra-special p-group. This does not work for p = 2 or p = 3, but for p = 5 we found the following example, which is in fact the only example of coclass 0 with K soluble that we have found so far.

Example 1 Let H be a split extension of an extra special group E of order 5^{3} and exponent 5 by SL(2, 5), with the natural action of SL(2, 5) on E/E′. This is the group A_{5}2^{1}◇5◇(3, 3) in the notation of [HoP 89]. More precisely, H = ⟨a, b, x, y, z|a^{4} = b^{3} = (ab)^{5} = [a^{2}, b] = 1, x^{5} = y^{5} = z^{5} = [x, z] = [y, z] = 1, [x, y] = z, a^{−1}xa = yz^{2}, a^{−1}ya = x^{−1}, b^{−1}xb = y^{−1}, b^{−1}yb = xy^{−1}⟩.

L is an irreducible 20-dimensional ZZ_{5}H-lattice, which is defined by the matrices below for a, b, x and y, which we have printed as 5-dimensional matrices over ZZ[ζ], where ζ is a 5^{th} root of 1:

a = ( \begin{matrix}−ζ & −ζ − ζ^{3} & −ζ^{3} & 0 & −1 − ζ^{3} \\ ζ − ζ^{2} + ζ^{3} & 1 + ζ^{3} & 2 + ζ + 2ζ^{2} + 2ζ^{3} & 1 + ζ^{3} & −ζ \\ 1 − ζ + ζ^{2} & −1 − ζ − 2ζ^{3} & −2 − ζ − 2ζ^{3} & −1 − ζ − ζ^{3} & −1 − ζ^{2} − ζ^{3} \\ −2 − ζ^{2} − ζ^{3} & ζ + ζ^{3} & −ζ^{2} + ζ^{3} & ζ + ζ^{3} & 1 + ζ + ζ^{3} \\ −ζ & −1 − ζ^{3} & −1 − ζ − ζ^{2} − 2ζ^{3} & −1 − ζ − ζ^{3} & ζ \end{matrix} )

b = ( \begin{matrix}−1 − ζ^{2} & −ζ & 2 + ζ + ζ^{2} + ζ^{3} & 1 + ζ^{3} & −1 − ζ^{3} \\ −1 − ζ − ζ^{3} & ζ − ζ^{2} + ζ^{3} & −1 + ζ − ζ^{2} + ζ^{3} & −1 + ζ − ζ^{2} & 1 + ζ^{2} + ζ^{3} \\ ζ − ζ^{2} + ζ^{3} & 1 − ζ + ζ^{2} & 2 + 2ζ^{2} + ζ^{3} & 2 + ζ^{2} + 2ζ^{3} & −2 − ζ − ζ^{2} − ζ^{3} \\ 1 + ζ^{2} & −2 − ζ^{2} − ζ^{3} & −2 − ζ − ζ^{2} − 2ζ^{3} & −2 − ζ − 2ζ^{3} & −ζ^{2} + ζ^{3} \\ 1 + ζ + ζ^{3} & −ζ & 1 + ζ^{2} & 1 − ζ & 0 \end{matrix} )

x = ( \begin{matrix}0 & 1 + ζ + ζ^{2} + ζ^{3} & 0 & 0 & 2 + ζ + ζ^{2} + ζ^{3} \\ ζ & −2 − ζ^{2} & 0 & 0 & −1 + ζ − ζ^{2} + ζ^{3} \\ 0 & 2 + ζ^{2} + 2ζ^{3} & 1 & 0 & 2 + 2ζ^{2} + ζ^{3} \\ 0 & −1 − ζ − ζ^{2} − 2ζ^{3} & 0 & 0 & −2 − ζ − ζ^{2} − 2ζ^{3} \\ 0 & 2 + ζ + ζ^{2} + ζ^{3} & 0 & ζ & 1 + ζ^{2} \end{matrix} )

y = ( \begin{matrix}ζ^{3} & 0 & −1 − ζ − ζ^{2} − ζ^{3} & 0 & 2 + ζ + ζ^{2} + ζ^{3} \\ 0 & −1 − ζ − ζ^{2} − ζ^{3} & −ζ^{3} & 0 & −1 + ζ − ζ^{2} + ζ^{3} \\ 0 & 0 & ζ & 0 & 2 + 2ζ^{2} + ζ^{3} \\ 0 & 0 & 1 + ζ + ζ^{2} + ζ^{3} & 0 & −2 − ζ − ζ^{2} − 2ζ^{3} \\ 0 & 0 & ζ^{3} & 1 & 1 + ζ^{2} \end{matrix} ).

The non-split extension X of L by H, which has coclass 0, is generated as a profinite group by the 6-dimensional matrices

( \begin{matrix}a & 0 \\ v_{a} & 1 \end{matrix} ), ( \begin{matrix}b & 0 \\ v_{b} & 1 \end{matrix} ), ( \begin{matrix}x & 0 \\ v_{x} & 1 \end{matrix} ), ( \begin{matrix}y & 0 \\ v_{y} & 1 \end{matrix} ),

where v_{a} = (4, 3, 0, 2, 0)
v_{b} = (0, 1, 0, 0, 4)
v_{x} = (0, 3, 0, 2, 0)
v_{y} = (0, 0, 2, 2, 0)

Then X ⊆ ZZ_{5}[ζ]^{5} >⊲H with X ∩ (ZZ_{5}[ζ]^{5}) = (1 − ζ)ZZ_{5}[ζ]^{5}. It is routine to check (by computer) that X has the required properties.

This example does not appear to generalize in any obvious way (such as to SL(2, p)). For the case in which K is not soluble, a sensible restriction to impose might be to assume that K is p-adic analytic, as defined in [Laz 65]. As pointed out in Definition 2.3.15 of [HoP 89], this is equivalent to the factors Φ^{i}(K)/Φ^{i+1}(K) of the Frattini series of K having bounded order. Examples of coclass 0 are not difficult to find.

Example 2 Let X = SL(3, ZZ_{2}), with ZZ_{2} the ring of 2-adic integers and let

Γ_{i} = {x ∈ X | x ≡ I(mod2^{i})}

Then K = Γ_{1} is a pro-2-group, and G = X/Z ≅ L_{3}(2). This example was also considered in Section 5 of [HoP 86], where it was shown that [Γ_{i}, Γ_{j}] = Γ_{i+j} for all i, j ≥ 1 (and so, in particular, γ_{i}(K) = Γ_{i}), and that, for all i ≥ 1, Γ_{i}/Γ_{i+1} is the irreducible 8-dimensional (Steinberg) representation of G. Thus cocl_{2}(X) = 0. It is clear that that K is nonsoluble and 2-adic analytic.

This example can be generalized as follows.

Proposition 2 Let G = L(IF_{q}) be a finite Chevalley group over the field IF_{q} defined over a simple Lie Algebra L , with q a power p. Assume that IF_{q} ⊗ L is simple as a IF_{q}Gmodule, let R be a discrete valuation ring with residue class field IF_{q}, and let X = L(R). Then cocl_{p}(X) = 0.

Proof: We take our definition of L(R) from Section 27 of [Hum 72], which is valid for all commutative rings (though only stated for fields). Let M be an admissible lattice of the Kostant ZZ-form U(L)_{ZZ} of the universal envelopping algebra of L. For a commutative ring S, define L(S) as the subgroup of Aut_{S}(S ⊗_{ZZ} M), as in the above reference, via the 1-parameter subgroups. Let π ∈ R generate rad(R), so that R/(π) = IF_{q}. Then, by definition, the ring epimorphism R → R/(π)^{i} induces a group epimorphism L(R) → L(R/(π)^{i}). Let K_{i} be the kernel of this map for i = 1, 2, …, so K_{1} = K and X/K ≅ G as required. We claim that K_{i}/K_{i+1} is isomorphic, as a G-module, to IF_{q} ⊗L. Let (IF_{q} ⊗L)_{M} be the image of IF_{q}⊗L in End_{IF_{q}}(IF_{q}⊗L) via the action of U(L)_{M}. Define ϕ_{i} : K_{i} → (IF_{q}⊗ L)_{M} by ϕ_{i}(1 + π^{i}x) = 1 ⊗_{IF_{q}} x ≡ x(mod(π)) where x ∈ End_{R}(R ⊗ M). The polynomials (over ZZ or ZZ/pZZ) describing L(R) as the subgroup of R-valued points of a split algebraic group show, upon differentiation, that ϕ_{i} maps K_{i} not only into End_{IF_{q}}(IF_{q} ⊗ M) but into (IF_{q} ⊗ L)_{M} (cf. [Ser 65], LA 1.4). Clearly ϕ_{i} is a nonzero homomorphism with kernel K_{i+1}, inducing a G-module homomorphism K_{i}/K_{i+1} → IF_{q} ⊗ L. Since we are assuming that IF_{q} ⊗ L is irreducible, ϕ_{i} is an isomorphism.

Clearly [K_{i}, K_{j}] ⊆ K_{i+j}, and a routine calculation in SL_{2} shows that [K_{i}, K_{j}] ⊈

K_{i+j+1}, (use [( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ b & 1 \end{matrix} )] = ( \begin{matrix}1 + ab + a^{2}b^{2} & a^{2}b \\ −ab^{2} & 1 − ab \end{matrix} ) ) and hence [K_{i}, K_{j}] =

K_{i+j} by irreducibility, and so K_{i} = γ_{i}(K) and G has coclass 0. q.e.d.

In [His 84] the Chevalley algebras L and finite fields IF_{q} are classified such that IF_{q} ⊗L is irreducible as IF_{q}L(IF_{q})-module. It seems that usually IF_{q} ⊗ L is also irreducible as IF_{p}L(IF_{q})-module when p is a prime with q = p^{α}. Typical examples are A_{p} for p| n̸ + 1, G_{2} for p ≠ 3, and most other simple Lie algebras for p ≠ 2.

In conclusion we suggest the study of the following two functions on pairs (G, p) with G a finite group, p a prime such that O_{p}(G) = 1. mcs(G, p) = smallest d such that there exists a soluble pro-p-group K and group X with K ⊲ X, X/K ≅ G and cocl_{p}(X) = d, and mcns(G, p) = smallest d such that there exists a nonsoluble pro-p-group K and group X with K ⊲ X, X/K ≅ G and cocl_{p}(X) = d. In particular the question of pairs (G, p) with mcs(G, p) = 0 or mcns(G, p) = 0 looks very interesting.

References

[DSMS 91] J.D. Dixon, M.P.F. du Sautay, A. Mann and D. Segal, ”Analytic pro-p-groups”, L.M.S. Lecture Note Series 157, C.U.P. 1991. [His 84] G. Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen, Arch. Math. 42 (1984), 408-416. [Hol 89] D.F. Holt, Enumerating perfect groups, J. Lon. Math. Soc. 39 (1989), 67-78. [HoP 86] D.F. Holt and W. Plesken, A_{5}-invariant 2-groups with no trivial sections,Quart. J. Math (Oxford) 37 (1986), 39-47. [HoP 89] D.F. Holt and W. Plesken, ”Perfect Groups”, Oxford University Press, 1989. [Hum 72] J.E. Humphreys, ”Introduction to Lie Algebras and Representation Theory”, Springer, 1972. [LeN 80] C.R. Leedham-Green and M.F. Newman, Space groups and groups of prime power order I, Archiv der Mathematik 35 (1980), 193-202. [Don 87] S. Donkin, Space groups and groups of prime power order VIII: Pro-p-groups of finite coclass and p-adic Lie algebras, J. Alg. 111 (1987), 316-342. [Laz 65] M. Lazard, Groupes analytiques p-adiques, IHEs Publ. Math. 26 (1965), 389-603. [Ser 65] J.P. Serre, ”Lie Algebras and Lie Groups”, W.A. Benjamin Inc., 1965.