EXTENSIONS OF PERIODIC LINEAR GROUPS WITH FINITE UNIPOTENT RADICAL

Richard E. Phillips† Julianne G. Rainbolt^{1} Jonathan I. Hall^{2} Ulrich Meierfrankenfeld^{3}

1Department of Mathematics, Saint Louis University, Saint Louis, MO 63103

email: rainbolt@slu.edu

2Department of Mathematics, Michigan State University, East Lansing, MI

48824 email: jhall@math.msu.edu

3Department of Mathematics, Michigan State University, East Lansing, MI

48824 email: meier@math.msu.edu

Dedication

This article is published posthumously for the author Richard Phillips. The other authors dedicate this paper to his memory.

Abstract

A group is called p-linear if it is isomorphic to a subgroup of GL(n, K) for some field K of characteristic p and some integer n. Let H be a normal subgroup of G and assume that both H and G/H are periodic and p-linear. In addition, assume that both H and G/H have finite unipotent radicals and that the Hirsch-Plotkin radical of H is C̆ernikov. The main result of this article is a proof that under these assumptions G is p-linear. An example is provided showing the result is false if the assumption regarding the Hirsch-Plotkin radical is removed.

1. Results

For finite n, the group G is said to be K-linear of degree n if it is isomorphic to a subgroup of GL(n, K) and is p-linear if it is K-linear for some field K of characteristic p.

The following theorem was proven as [1, Theorem A].

( 1.1) Theorem. Let G be a periodic subgroup of GL(n, K) with trivial unipotent radical. Then for every normal subgroup H of G, the image G/H is Klinear of degree bounded by a function of n.

Here the statement on bounded degree indicates that there is a positive, nondecreasing function f such that the degree of G/H is always less than or equal to f(n).

Wehrfritz [2] gave a second proof of the theorem and was able to weaken the hypotheses by assuming only that H has trivial unipotent radical. A slightly different proof of the theorem is given below (one that in fact includes groups G with finite unipotent radical, as remarked below).

The main concern here is proving a converse result to this theorem. It is not in general the case that an extension of a linear group by a linear group is linear. It is also not clear what shape the best converse to the theorem would take.

Consider the following:

Hypothesis (∗)

G is a locally finite group and M a normal subgroup of G satisfying:

(i) M is a central product A Y L_{1} Y ⋯ Y L_{i} Y ⋯ Y L_{s}, for some s; (ii) each L_{i} is a quasisimple group of Lie type in characteristic p; (iii) A is a periodic abelian p′-group of finite rank r; (iv) G/M is finite.

The rank of a periodic abelian group B is the largest m for which there is a prime q and an elementary abelian q-subgroup C of B having order q^{m}. If p = 0, then s = 0 in (i) and A is periodic abelian of finite rank in (iii).

Each of the groups L_{i} in (i) has, naturally defined, a Lie rank r_{i} and a field of definition K_{i} ≤ \overline{𝔽}_{p}. (See [1, Lemma 9(iv)] or Theorem 2.5 below.) (In characteristic p = 0 we set 𝔽_{p} = ℚ hence \overline{𝔽}_{p} = \overline{ℚ}.) We define the M-rank of G with (∗) to be r +∑_{i=1}^{s} r_{i}. We define the field K^{M} to be the smallest subfield of \overline{𝔽}_{p} that contains each K_{i} and all q^{k}th roots of unity, for every k and all primes q for which A contains a subgroup ℤ_{q^{∞}}. Thus M is K^{M}-linear.

( 1.2) Theorem. If G is a periodic K-linear group of degree n with finite unipotent radical U(G), then G has a characteristic subgroup M for which (G, M) satisfies (∗), where p = Char(K). In this case, M can be chosen so that the M-rank of G is bounded by a function of n and the index |G/M| is bounded by functions of n and |U(G)|.

( 1.3) Theorem. A group G with (∗) is K^{M}-linear of degree bounded by a function of its M-rank and the index |G/M|.

As a corollary, we get Winter’s theorem [3]. The usual proofs are more elementary than this; see [4, Theorem 1.L.2] and [5, Theorem 9.5].

( 1.4) Corollary. If G is a periodic K-linear group with trivial unipotent radical, then G is \overline{𝔽}_{p}-linear, where p is the characteristic of K. In particular, G is countable.

Since uncountable unipotent linear groups of exponent p exist, Winter’s theorem is false without some restriction on the radical. Nevertheless, because of it we can restrict (most) discussion to the question of whether or not a given group is p-linear (that is, linear over some field of characteristic p).

We have a partial converse to Theorem 1.1.

( 1.5) Theorem. Let H be a normal subgroup of G and assume that (a) G/H is a periodic p-linear group with finite unipotent radical;

(b) H is a periodic p-linear group with finite unipotent radical;

(c) the Hirsch-Plotkin radical of H is C̆ernikov. Then G is p-linear.

A C̆ernikov group is one that satisfies the minimal condition on subgroups and has an abelian subgroup of finite index. (See [4, pp. 28–38]. By [5, Corollary 9.8], in linear groups the second condition is a consequence of the first.) The Hirsch-Plotkin radical of a group is its largest locally nilpotent normal subgroup.

The next section contains proofs of Theorems 1.1-1.5. The final section contains examples. The first example shows that Theorem 1.5 is false under (a) and (b) alone. The other two examples demonstrate the impossibility in Theorem 1.5 of bounding the representing degree of G in terms of deg G/H and deg H, indeed in terms of the isomorphism class of one of these and the degree of the other.

The problem discussed in this article was one that Richard Phillips was working on during the last years of his life. He wrote an initial draft of this article in conjunction with Julianne Rainbolt. After Richard Phillips’ death, Jonathan Hall, Ulrich Meierfrankenfeld, and Julianne Rainbolt completed the revisions of the article. The authors thank Felix Leinen and the referees for helpful remarks on earlier drafts of this article.

2. Proofs

Let G be a locally finite group and π be any set of primes. Then O_{π}(G) denotes the largest normal subgroup of G all of whose elements are π-elements. If G is periodic and linear in characteristic p, then O_{p}(G) is the unipotent radical of G. In the following, E(G) denotes the subgroup of G generated by the components of G, where a component of G is a subnormal quasisimple subgroup.

( 2.1) Theorem. (Schur, [4, Theorem 1.L.1]) Periodic linear groups are locally finite.

The next two lemmas are elementary.

( 2.2) Lemma. If G is K_{0}-linear of degree n and K is a subfield of K_{0} such that |K_{0} : K| is finite, then G is K-linear of degree n|K_{0}: K|.

( 2.3) Lemma. If M is K-linear of degree m and has finite index in G, then G is K-linear of degree m|G: M| via the induced representation.

( 2.4) Lemma. Let H be K-linear and Z a finite normal subgroup of the center Z(H). Then H/Z is K-linear of degree bounded by a function of the degree of H and |Z|.

In particular, if the factors H_{i} are K-linear, then a central product H_{1} Y H_{2} = (H_{1} × H_{2})/Z over a finite central subgroup Z is K-linear of degree bounded by a function of the degrees of the H_{i} and |Z|.

Proof. As the direct product of two K-linear groups is K-linear with degree equal to the sum of the two degrees, this an immediate consequence of [1, Proposition 3(ii)].

A proof of the following theorem can be found at [6, Lemmas 15.6, 15.10, and Theorem 15.12]. It is very similar to [1, Proposition A] and [7, 1.2].

( 2.5) Theorem. Let G be a periodic linear group of degree n over a field in characteristic p and having trivial unipotent radical.

Let { L_{i} | 1 ≤ i ≤ t } consist of all components of G that have Lie type in characteristic p. The central product E_{L,p}(G) = L_{1} Y ⋯ Y L_{t} is characteristic in G, and t ≤ n/2. Furthermore, G has a characteristic abelian subgroup A such that the subgroup M(G) = A Y E_{L,p}(G) is characteristic in G and has finite index bounded by a function of n.

Proof of Theorem 1.2.

G is locally finite by Theorem 2.1.

As the unipotent radical U = U(G) is finite, C_{G}(U) is a characterisitic subgroup of G of finite index bounded by a function of |U|. Therefore we may assume that U ≤ Z(G).

Let Ḡ = G/U. Note that Ḡ is a K-linear group of degree n with trivial unipotent radical.

Let N̄ = M(Ḡ) be the subgroup Ā Y L̄_{1} Y ⋯ Y L̄_{s} of Theorem 2.5. Let L_{i} be the derived group of the preimage of L̄_{i} in G, A the p′-part of the preimage of Ā, and M = A Y L_{1} Y ⋯ Y L_{s}. Then M is characteristic in G, and the pair (G, M) satisfies (∗).

The various rank(L_{i}) are bounded by a function of n [1, Lemma 9(b)(i)]. Since A is p′ and periodic, it has finite rank bounded by a function of n by Maschke’s Theorem.

Proof of Theorem 1.3.

By (iv) and Lemma 2.3, we may assume G = M. Each L_{i} is K^{M}-linear by assumption and has a center that is finite of order bounded by a function of the Lie rank of L_{i} ([1, Lemma 10(viii)] or Lemma 2.7). Thus E(G) = L_{1} Y ⋯ Y L_{s} is K^{M}-linear by Lemma 2.4 and has finite center of order bounded in terms of the Lie ranks of the L_{i}. By definition, (K^{M})^{×} contains a copy of ℤ_{q^{∞}} whenever A has infinite q-part, so A is K^{M}-linear of degree equal to its rank r. A second application of Lemma 2.4 then proves that G = M = A Y E(G) is K^{M}-linear, as desired. Furthermore, its degree is controlled as described.

Proof of Corollary 1.4.

As G is periodic and K-linear, it has (∗) by Theorem 1.2 and so is \overline{𝔽}_{p}-linear by Theorem 1.3.

Proof of Theorem 1.1.

By Theorem 1.2, G has (∗) for a normal subgroup M, and the M-rank of G and |G/M| are both bounded by functions of n. In Ḡ = G/H, set M̄ = MH/H. Then the pair (Ḡ, M̄) inherits (∗), so by Theorem 1.3 Ḡ is linear of degree bounded in terms of the M̄-rank of Ḡ (at most the M-rank of G) and |Ḡ: M̄| (≤ |G: M|). Thus the degree of G/H is bounded by a function of n, as desired.

Remark. The same proof actually gives something slightly stronger than Theorem 1.1. We need only require that the unipotent radical of G be finite, in which case the representation degree of G/H is bounded by a function of n and of the order of the radical.

( 2.6) Lemma. Let B be a class 2 nilpotent periodic group with B′ ≤ H ≤ Z(B). Assume that B/H has finite rank and that H is C̆ernikov. Then B has an abelian subgroup of finite index and finite rank.

Proof. The central subgroup H is C̆ernikov and so has only finitely many prime divisors; let the set of these be π. The set of π′-elements of B then forms a characteristic abelian subgroup O_{π′}(B) ≃ O_{π′}(B/H) of finite rank. Let B = O_{π′}(B) × B_{0}, where B_{0} = O_{π}(B) consists of all π-elements of B. B_{0}/H is of finite rank and is, in fact, C̆ernikov since π is finite. But then B_{0} is C̆ernikov by [4, Lemma 1.E.7]. In particular it contains an abelian subgroup A_{0} of finite rank and of finite index in B_{0}. The subgroup O_{π′}(B) × A_{0} is then abelian of finite rank and has finite index in B, as desired.

( 2.7) Lemma. Let periodic N have a central subgroup H such that N̄ = N/H is a central product M̄_{1} Y ⋯ Y M̄_{t} of finitely many infinite quasisimple groups M̄_{i} of Lie type. Then N = H Y E(N), where E(N) = N_{1} Y ⋯ Y N_{t} has finite center and is a central product of quasisimple groups N_{i} of Lie type with N̄_{i} = M̄_{i}.

Proof. By the Three Subgroups Lemma, Z_{2}(N) = Z(N). Also N′ = [N′H, N′H] ≤ N′′; so N′ is perfect, and N = H Y N′. Indeed N/H = N′H/H ≃ N′/N′ ∩ H has image N′/Z(N′). Therefore N′ = E(N), the central product of the components N_{i} = (M_{i})′, the derived subgroups of the preimages M_{i} of the various M̄_{i}.

Each simple periodic infinite Lie type group L is a direct limit of finite simple groups of the same Lie type, and so any element of its multiplier occurs already within the multiplier of some finite subgroup of Lie type. Exceptional multipliers for finite Lie type groups occur only over small fields, and the canonicial multipliers come from the natural or spin representations of fixed degree bounded in terms of the Lie rank [8]. Thus the multiplier of L is finite of order bounded by the rank of L and comes from a representation of degree also bounded by the rank. (See also [1, Lemma 10].)

We next prove a special case of Theorem 1.5.

( 2.8) Theorem. Let H be a normal subgroup of G and assume that (a) G/H is a periodic p-linear group with finite unipotent radical;

(b) H is finite. Then G is p-linear with finite unipotent radical.

Proof. By Theorem 2.1 G/H is locally finite, so G itself is locally finite (by [4, Lemma 1.A.2]).

Since the quotient of any linear group by its unipotent radical is also linear, we may replace H by the preimage in G of the unipotent radical of G/H. Thus we may assume that G/H has trivial unipotent radical.

As H is finite, C_{G}(H) has finite index in G. Replacing H by H ∩ C_{G}(H) and G by C_{G}(H), we may by Lemma 2.3 assume that H ≤ Z(G).

By Theorem 1.2, M(Q) has finite index in Q, and so its preimage in G has finite index. Thus by Lemma 2.3 in proving G to be p-linear we can assume Q = M(Q) = A Y E(Q) = A Y L_{1} Y ⋯ Y L_{s}. Indeed again by Lemma 2.3 we just need to show that the preimage G_{0} of A Y E_{∞}(Q) is p-linear, where E_{∞}(Q) = L_{1} Y ⋯ Y L_{t} only includes those components L_{i} of Q that are infinite.

By Lemma 2.7, G_{0} = B Y E(G_{0}), where E(G_{0}) is a central product of Lie type groups and B(≥ H) has class at most 2 with B/H = A. By Lemma 2.6, B has an abelian subgroup B_{0} of finite index and finite rank. As O_{p}(G) ≤ H is finite, B_{0} in turn has a finite index abelian p′-subgroup A_{0} of finite rank. Therefore G_{0} ≥ A_{0} Y E(G_{0}) is p-linear by Theorem 1.3, as desired.

Proof of Theorem 1.5.

By Theorem 2.1 both H and G/H are locally finite, so G itself is locally finite (by [4, Lemma 1.A.2]). By Theorem 2.8 we may assume that O_{p}(H), the unipotent radical of H, is trivial.

By Theorem 2.5 M(H) is characteristic of finite index in H, so by Theorem 2.8 we may assume H = M(H) = C Y E(H), where C(≥ Z(E(H))) is abelian, characteristic in H, and C̆ernikov. By [1, Lemma 9] C_{G}(E(H))E(H) has finite index in G, so we may take G = C_{G}(E(H))E(H) (by Lemma 2.3 again). By [4, Theorem 1.F.3], any periodic subgroup of Aut(C) is finite, so we may further assume G = M(H)C_{G}(M(H)). Thus G = C_{G}(M(H)) Y C Y E(H), a central product (indeed C ≤ C_{G}(M(H))). E(H) has finite center by [1, Lemma 10] (or Lemma 2.7). As E(H) (≤ H) is p-linear by hypothesis, when we have proven that C_{G}(M(H)) Y C = C_{G}(M(H)) is p-linear we will be done by Lemma 2.4. Linear G/H = M(H).C_{G}(M(H))/H ≃ C_{G}(M(H))/H ∩ C_{G}(M(H)) = C_{G}(M(H))/C, so we are reduced to the case C = H ≤ Z(G).

As O_{p}(H) = 1, O_{p}(G) is isomorphic to the unipotent radical of Q = G/H and is, in particular, finite. Thus by Theorem 2.8 we may assume that O_{p}(G) and O_{p}(Q) are trivial. We now complete the proof as in the last two paragraphs of the proof of Theorem 2.8.

3. Examples

( 3.1) Lemma. There is an infinite sequence of primes

q_{1} < p_{1} < ⋯ < q_{i} < p_{i} < ⋯

for i ∈ ℤ^{+}, with q_{i}| p_{i} − 1.

Proof. The proof is by induction on i. Let q_{1} be any prime number. A theorem of Dirichlet [9, p. 250] asserts that the sequence {1 + kq_{1} | k = 1, 2, …} contains an infinite number of primes. Let p_{1} be any such prime. Then q_{1}|(p_{1} − 1). Now let q_{2} be a prime greater than p_{1}.

Suppose that we have chosen q_{1}, …, q_{s+1} and p_{1}, …, p_{s} such that

q_{1} < p_{1} < ⋯ < q_{s} < p_{s} < q_{s+1}

Now choose the prime p_{s+1} in the sequence {1 + kq_{s+1} | k = 1, 2, …} and the prime q_{s+2} with q_{s+2} > p_{s+1}.

( 3.2) Example.

We first show that Theorem 1.5 is false when only (a) and (b) are assumed (even when (a) and (b) are strengthened to require that the unipotent radicals be trivial).

For p_{i}, q_{i} (i ∈ ℤ^{+}) as in Lemma 3.1, let S_{i} be cyclic of order p_{i} and R_{i} cyclic of order q_{i}. Let F_{i} = S_{i}.R_{i} be the Frobenius group of order p_{i}q_{i}. Then F^{k} = ⊕ ∑_{i=1}^{k} F_{i} has representation degree at least 2k in each characteristic, even though it is the extension S^{k}.R^{k} of cyclic S^{k} = ⊕ ∑_{i=1}^{k} S_{i} of degree 1 over any prime not in the sequence (and degree at most 2 in general) by cyclic R^{k} = ⊕ ∑_{i=1}^{k} R_{i} also of degree 1 over any prime not in the sequence (and degree at most 2 in general).

Thus F^{∞} = ⊕ ∑_{i} F_{i} = lim_{k} F^{k} is not linear in any characteristic even though it is the extension of rank 1 linear S^{∞} = ⊕ ∑_{i} S_{i} = lim_{k} S^{k} by rank 1 linear R^{∞} = ⊕ ∑_{i} R_{i} = lim_{k} R^{k}. The Hirsch-Plotkin radical of F^{∞} is S^{∞} and is not C̆ernikov.

( 3.3) Example.

The groups F^{k} of the first example show that the degree of linear G can not be bounded in terms of the degree, deg H, of the normal subgroup H and deg G/H. We can refine the example to show that knowledge of deg H and the isomorphism type of G/H is still not enough in general to bound the degree of linear G.

Indeed, let linear F^{(k)} = F^{k} ⊕ ∑_{i>k} R_{i} be the split extension of S^{k} (as before) by R^{∞} (with the R_{i}, for i > k, acting trivially). As F^{k} ≤ F^{(k)}, the representation degree still goes to infinity, but now F^{(k)}/S^{k} ≃ R^{∞}, for all k.

( 3.4) Example.

Finally, as a complement to the previous example, we show that the degree of linear G can not in general be bounded in terms of deg G/H and the isomorphism type of H.

Fix a characteristic p and a prime q ≠ p. Let, for n ∈ ℤ^{+},

E(n) = ⟨x, y, z | x^{q^{n}} = y^{q^{n}} = z^{q^{n}} = 1, [x, y] = z, [x, z] = [y, z] = 1 ⟩ .

Let E^{∗}(n) be the central product of E(n) and ℤ_{q^{∞}} with z identified with an element of order q^{n} of ℤ_{q^{∞}}. Then H = ℤ_{q^{∞}} is p-linear of degree 1 and E^{∗}(n)/H ≃ Z_{q^{n}} × Z_{q^{n}} is p-linear of degree 2.

We claim that the degree of E(n) (and hence of E^{∗}(n)) in characteristic p is q^{n}. Abelian W = ⟨x, z⟩ has index q^{n}, so this is an upper bound on the degree. Consider the restriction of a faithful representation to W. There is some degree 1 constituent ζ with W = ⟨w, z⟩ = C_{E(n)}(w), where ⟨w⟩ = ker(ζ). Then by Clifford’s Theorem the ζ^{y^{i}}, for i = 0, …, q^{n} − 1, are distinct constituents of the restriction.

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