Isomorphism questions for metric ultraproducts of finite quasisimple groups

1 Introduction

In [5, 6], Thom and Wilson discussed various properties of metric ultraproducts of finite simple groups. In particular, they asked which such ultraproducts can be isomorphic. In [5, Theorem 2.2], a metric ultraproduct of alternating groups is distinguished from a metric ultraproduct of classical groups of Lie type, where the permutation degrees resp. dimensions of the natural module tend to infinity. This is done by considering the structure of centralizers of torsion elements in these groups (see [5, Theorems 2.8 and 2.9]). In the case of a metric ultraproduct of classical groups of Lie type, in [5, Theorem 2.8], investigating the structure of such centralizers of semisimple and unipotent torsion elements, Thom and Wilson even extract the “limit characteristic” of the group. At the end of [5, Section 2], they ask which metric ultraproducts of classical groups of different types can be isomorphic.

In this note, we will give an almost complete answer to this question in the case when the field sizes are bounded. We will show that, for a metric ultraproduct of alternating or classical groups of Lie type of unbounded rank over fields of bounded size, one can extract the Lie type (up to one exception). Also one can extract the “limit field size”. Our results are summed up in Theorem 1 below. To state it, we first need to introduce some notation which we fix for the whole article.

Let ℋ=(Hi)i∈I be a sequence of groups where either Hi=Sni is a symmetric group or Hi=Xi⁢(qi) is a classical group of Lie type Xi over the finite field 𝔽qi with qi elements (resp. 𝔽qi2 in the unitary case, i∈I). In the latter case, we let each Xi be one of GLni, Sp2⁢mi, GO2⁢mi±, GO2⁢mi+1 (qi odd), or GUni for suitable mi,ni∈ℤ+ (i∈I).

Recall that a normℓ on a group H is a function H→[0,∞] such that ℓ⁢(h)=0 iff h=1H, ℓ⁢(h)=ℓ⁢(h-1)=ℓ⁢(hg), and ℓ⁢(g⁢h)≤ℓ⁢(g)+ℓ⁢(h) for all g,h∈H. Call a pair (H,ℓ), where H is a group and ℓ a norm on it, a normed group. Recall that the metric ultraproduct of a sequence of normed groups (Hi,ℓi)i∈I along an ultrafilter 𝒰 on the set I is defined as the quotient ∏i∈IHi/N, where N:={(hi)i∈I∈∏i∈IHi∣lim𝒰ℓi(hi)=0} is the normal subgroup of 𝒰-null sequences.

Throughout, let G:=ℋ𝒰met be the metric ultraproduct of the groups Hi from above equipped with the normalized Hamming norm

resp. the normalized rank normℓrk(g):=rk(1-g)/n when Hi is a symmetric resp. a classical linear group of Lie type along the ultrafilter 𝒰. Assume that the permutation degrees resp. dimensions of the natural module ni of Hi (i∈I) tend to infinity along 𝒰.

Note that, since 𝒰 is an ultrafilter, we may assume that each group Hi is of the same type, i.e., all groups Hi are either symmetric, linear, symplectic, orthogonal, or unitary groups. In these five distinct cases, we write S𝒰, GL𝒰, Sp𝒰, GO𝒰, or GU𝒰 for G. Also, when the field sizes qi are bounded, we may assume that qi=q is constant (i∈I), setting q:=lim𝒰qi. Throughout the article, set Z:=𝐙(G) to be the center of G and G¯:=G/Z.

If the groups Hi (i∈I) are symmetric groups, then Z=𝟏 and G¯=G. Now assume that all groups Hi are of type X⁢(qi) (i∈I; i.e., they are not symmetric groups). Then G¯=G/Z=ℋ¯𝒰met is the metric ultraproduct of the groups H¯i:=Hi/𝐙(Hi) with respect to the projective rank norm

which is defined on the general projective linear group. By the results from [3], G¯ is the unique simple quotient of G.

Similarly to the above, write PGL𝒰, PSp𝒰, PGO𝒰, or PGU𝒰 for G¯ when all the groups Hi (i∈I) are linear, symplectic, orthogonal, or unitary groups. Moreover, in this case, if all the fields 𝔽qi (i∈I) are equal to 𝔽q (or 𝔽q2 in the unitary case), we write GL𝒰⁡(q), Sp𝒰⁡(q), GO𝒰⁡(q), GU𝒰⁡(q) resp. PGL𝒰⁡(q), PSp𝒰⁡(q), PGO𝒰⁡(q), PGU𝒰⁡(q) for G resp. G¯. Write GL𝒰⁡(k) resp. PGL𝒰⁡(k) for the metric ultraproduct of groups GLni⁡(k) resp. PGLni⁡(k) over the field k (i∈I) along 𝒰. Write 𝐌n⁢(k) for the matrix ring of degree n over the field k and P⁢𝐌n⁢(k) for the associated projective space (𝐌n⁢(k)∖{0})/k×. Set 𝐌n(q):=𝐌n(𝔽q) and P𝐌n(q):=P𝐌n(𝔽q). Also write 𝐌𝒰, 𝐌𝒰⁢(q) resp. P⁢𝐌𝒰, P⁢𝐌𝒰⁢(q) for the metric ultraproduct of the spaces 𝐌ni⁢(qi), 𝐌ni⁢(q) resp. P⁢𝐌ni⁢(qi), P⁢𝐌ni⁢(q) with respect to the metrics

so that GL𝒰⊆𝐌𝒰, GL𝒰⁡(q)⊆𝐌𝒰⁢(q), PGL𝒰⊆P⁢𝐌𝒰, PGL𝒰⁡(q)⊆P⁢𝐌𝒰⁢(q).

The main result of this article is now as follows.

Let us conclude this introduction by saying some words about the proof of Theorem 1. Our strategy is to compute double centralizers of semisimple torsion elements of a fixed order o∈ℤ+ in the above metric ultraproducts. If the sizes qi (i∈I) of the fields 𝔽qi are bounded, it turns out that these are always finite abelian groups. Then we consider the maximal possible exponent which such a double centralizer can have. It turns out that this data is enough to determine the limit field size q and the Lie type (up to the exception mentioned in Theorem 1). If the field sizes qi (i∈I) tend to infinity, a double centralizer of such a torsion element of order o>2 is always infinite. This separates this case from the former one. Throughout the article, we will make frequent use of the exposition given in [4, Subsection 3.4.2, § 1]. This paragraph in the author’s PhD thesis provides a concise presentation of the classification of the possible indecomposable blocks of elements from classical groups of Lie type stabilizing a form (which is, of course, well known). For the convenience of the reader, we will quote the facts needed from it in the subsequent section.

2 Notation and auxiliary results from [4]

In this section, we introduce the necessary notation and quote the facts from [4] which we need for the proof of Theorem 1. Keep the notation from the introduction. Subsequently, G will be one of S𝒰, GL𝒰⁡(q), Sp𝒰⁡(q), GO𝒰⁡(q), or GU𝒰⁡(q) (so that G¯ is either S𝒰, PGL𝒰⁡(q), PSp𝒰⁡(q), PGO𝒰⁡(q), or PGU𝒰⁡(q), respectively). We refer to these five distinct cases as the symmetric case, the linear case, the symplectic case, the orthogonal case, and the unitary case, respectively. The last four cases are grouped together as the classical case. The symplectic and orthogonal case are grouped together as the bilinear case.

For the rest of this section, we assume that we are in the classical case. Then set k=𝔽q in the non-unitary case and k=𝔽q2 in the unitary case. Write p:=char(k) for the characteristic of the field k.

Subsequently, assume that we are not in the linear case. Then we let σ:k→k be the q-Frobenius map x↦xq, i.e., σ=idk in the bilinear case, whereas σ is the unique involution of k=𝔽q2 in the unitary case. For a monic polynomial

write χ∗=a0-σ⁢Xdeg⁡(χ)⁢χσ⁢(X-1) for the dual polynomial of χ. Here

is the polynomial χ with coefficients twisted by the automorphism σ. Call χ self-dual iff χ=χ∗. Furthermore, write fi for the sesquilinear forms stabilized by the group Hi (and Qi for the quadratic forms in the orthogonal case for p=2, i∈I). Note that a metric ultraproduct of groups GO2⁢mi+1⁡(q) for q a power of p=2 with respect to the projective rank norm where the mi tend to infinity along the chosen ultrafilter is isomorphic to such an ultraproduct of groups Sp2⁢mi⁡(q). Actually, GO2⁢mi+1⁡(q)≅Sp2⁢mi⁡(q), but the natural module of GO2⁢mi+1⁡(q) has one-dimensional radical rad⁡(fi), whereas rad⁡(fi)=0 in the second case. Hence we may and do assume that the forms fi (i∈I) are all non-singular and exclude groups of type GO2⁢m+1⁡(q) for q even. Write

for the Frobenius block associated to the monic polynomial

i.e., multiplication by X¯ in the quotient ring k⁢[X]/(χ) represented in the basis

Recall that, by the existence of the generalized Jordan normal (or primary rational canonical form) form, each linear transformation g∈𝐌n⁢(k) can be written as

where the cχ⁢(g) are uniquely determined. A polynomial χ∈k⁢[X] is here called primary iff it is the power ie (e∈ℤ+) of an irreducible polynomial i∈k⁢[X].

Here are the results needed from [4]. Subsequently, the group H is one of the groups Hi (i∈I), and the form f is alternating bilinear, symmetric bilinear, or σ-sesquilinear in the symplectic, orthogonal, and unitary case, respectively. Hence the sign ε of f is -1 in the symplectic case, and ε=+1 in the orthogonal and unitary case. The form Q is a compatible quadratic form in the orthogonal case for p=2, i.e., Q⁢(u+v)=Q⁢(u)+f⁢(u,v)+Q⁢(v) for all u,v. Recall that two sesquilinear forms f and f′ (or quadratic forms Q and Q′) on a vector space V are called linearly equivalent if f(u,v)=f′(u.g,v.g) (resp. Q(v)=Q′(v.g)) for all u,v∈V and some g∈GL⁡(V).

An indecomposable block of an element g from a classical group of Lie type stabilizing the sesquilinear form f with natural module V is a subspace W≠0 on which f is non-singular (i.e., W∩W⊥=0) and which is minimal with this property. Out of the theory of indecomposables presented in [4, Subsection 3.4.2, § 1], one can also deduce the following fact which we will use in the text.

The next result, which can also be deduced from [4, Subsection 3.4.2, § 1], describes the shape of f on a “semisimple” indecomposable block.

We will also make use of the relations presented in the next remark.

3 Description of conjugacy classes in S𝒰, GL𝒰⁡(q), and PGL𝒰⁡(q)

In this section, we give a description of the conjugacy classes of groups of type S𝒰 or PGL𝒰⁡(q). We will make use of this in the subsequent sections.

We start with some definitions. Write Sn=Sym⁡(n¯) for the symmetric group acting on the set n¯:={1,…,n}. Let Ck⁢(σ)⊆Sn (for a permutation σ∈Sn and k∈ℤ+) denote the set of all k-cycles of σ, and let ck(σ):=|Ck(σ)| denote the number of k-cycles of σ. Moreover, let Ωk⁢(σ) be the set on which all k-cycles of σ act. Set nk(σ):=|Ωk(σ)|=kck(σ) to be the cardinality of Ωk⁢(σ). Call the permutation σ∈Snk-isotypic when nk⁢(σ)=n, i.e., σ has only cycles of length k. Call σ isotypic if it is k-isotypic for one number k∈ℤ+. Similarly, for g∈𝐌n⁢(k) and a monic primary polynomial χ∈k⁢[X], let cχ⁢(g) be the number of Frobenius blocks F⁢(χ) in the generalized Jordan normal form of g (see Section 2). Let Vχ⁢(g) be the subspace on which g acts as F⁢(χ)⊕cχ⁢(g) with respect to this normal form, and set nχ(g):=dim(Vχ(g)). Clearly, Vχ⁢(g) is not uniquely determined as it depends on the decomposition of the vector space kn associated to the chosen generalized Jordan normal form of g. However, by writing Vχ⁢(g) we mean that these subspaces of kn are the decomposition of kn coming from such a normal form of g. Then, of course, nχ⁢(g)=deg⁡(χ)⁢cχ⁢(g). For a monic polynomial χ∈k⁢[X], say that g is χ-isotypic if g≅F⁢(χ)⊕c for some c∈ℤ+.

At first, we consider GL𝒰⁡(q) instead of PGL𝒰⁡(q). Subsequently, all polynomials from k⁢[X] that occur in the text are meant to be monic polynomials.

Conjugacy classes in S𝒰 and GL𝒰⁡(q). For an integer k∈ℤ+ and a polynomial ξ∈k⁢[X], define rk(σ):=|fix(σk)|/nresp.rξ(g):=dim(ker(ξ(g)))/n for σ∈Sn resp. g∈𝐌n⁢(q). Here fix(σ):={x∈n¯∣x.σ=x} is the set of fixed points of the permutation σ. Extend this definition to S𝒰 and 𝐌𝒰⁢(q) by setting

for σ=(σi)¯i∈I resp. g=(gi)¯i∈I. Both expressions are well-defined since, for σ=(σi)¯i∈I=(τi)¯i∈I∈S𝒰, one has

Similarly, if g=(gi)¯i∈I=(hi)¯i∈I∈GL𝒰⁡(q) and

we have

and the latter tends to zero along 𝒰. Here we used the same trick as in estimate (3.1) above to bound drk⁢(gij,hij) by j⁢drk⁢(gi,hi) (j=0,…,k) in estimate (3.2). Write r(σ):=(rk(σ))k∈ℤ+ and r(g):=(rξ(g))ξ∈k⁢[X]. Now define qk⁢(σ) for k∈ℤ+ via the equality

Write q(σ):=(qk(σ))k∈ℤ+. Applying Möbius inversion, we obtain

Alternatively, one can think of qk⁢(σ) as the 𝒰-limit of the normalized cardinality of the support of all k-cycles in σi (i∈I), i.e.,

Similarly, for χ=ie∈k⁢[X] primary, define qχ⁢(g) via the equality

for all polynomials ξ∈k⁢[X]. Here gcd⁡S is the greatest common divisor of the elements from S. Write q(g):=(qχ(g))χ⁢primary. Alternatively, one can think of qχ⁢(g) as the 𝒰-limit of the normalized dimensions of the (not unique) subspaces Vχ⁢(gi) (i∈I), i.e.,

where kχ=deg⁡(χ)=e⁢deg⁡(i). This is because, when g acts as F⁢(χ) and ξ∈k⁢[X], then dim⁡(ker⁡(ξ⁢(g)))=deg⁡(gcd⁡{χ,ξ}).

We claim that the conjugacy classes in S𝒰 resp. 𝐌𝒰⁢(q) are in one-to-one correspondence with all tuples (qk⁢(σ))k∈ℤ+ resp. (qχ⁢(g))χ⁢primary, where the only condition on the sequences are that

Here we let GL𝒰⁡(q) act on 𝐌𝒰⁢(q) by conjugation. The element g lies in GL𝒰⁡(q) iff qχ⁢(g)=0 for χ=Xe (e≥1). Indeed, one sees easily that r⁢(σ) resp. r⁢(g) is conjugacy invariant, and so is q⁢(σ) resp. q⁢(g) for σ∈S𝒰 and g∈𝐌𝒰⁢(q).

To see the converse, let

be elements such that q⁢(σ)=q⁢(τ) and q⁢(g)=q⁢(h), respectively.

Find a sequence (Ni)i∈I tending to infinity along 𝒰 such that

resp. such that

Then, by the triangle inequality,

resp.

Hence we can conjugate a big part of ⊔k=1NiΩk⁢(σi) equivariantly to a big part of ⊔k=1NiΩk⁢(τi) resp. an almost full-dimensional part of

equivariantly (with no error in the limit; here again Vχ⁢(gi) resp. Vχ⁢(hi) are not unique). The remaining part of σi and τi resp. gi and hi can be modified into one big cycle resp. one big Frobenius block with no change of σ and τ resp. g and h, since Ni→𝒰∞. Then we conjugate this cycle of σi resp. Frobenius block of gi onto the one of τi resp. hi.

The case of P⁢𝐌𝒰⁢(q). Let the group k×=𝔽q× act on all (monic) polynomials ξ∈k⁢[X] by ξ.z:=z-kξξ(zX), where kξ:=deg(ξ). Extend this action to all tuples q=(qχ)χ⁢primary with ∑χ⁢primaryqχ≤1 via

and denote by q¯ the orbit orbk×⁡(q) of q under this action of k×.

Suppose that G¯=PGL𝒰⁡(q). We claim that the conjugacy classes of elements g¯∈P⁢𝐌𝒰⁢(q) are classified by the bijection g¯G¯↦(qχ⁢(g))¯χ⁢primary, where g is any lift of g¯ in 𝐌𝒰⁢(q) (here we exclude the tuple q where qX=1 and qχ=0 otherwise).

Indeed, the map is well-defined since any other h such that h¯=g¯∈P⁢𝐌𝒰⁢(q) is of the form zg for some z∈k× (as k=𝔽q is finite) so that

Also q is constant on conjugacy classes of 𝐌𝒰⁢(q) (under the action of GL𝒰⁡(q)). Conversely, if qχ⁢(h)=qχ.z⁢(g)=qχ⁢(z⁢g) for some fixed z∈k× and all χ∈k⁢[X] primary, then, from the above, we derive that the elements g and z-1⁢h of 𝐌𝒰⁢(q) are conjugate so that g¯ and h¯ are conjugate in P⁢𝐌𝒰⁢(q). This proves the claim.