Research Summary
James B. Wilson
(Last updated March 14, 2013)
I study finite groups and develop algorithms to find
the structure of very large groups. This involves various tools, often outside of group theory: e.g. bilinear
maps, ∗-rings, Jordan and Lie algebras, varieties of
groups, Morita duality theory, and NP-complete sets.
Contents
1 Group decompositions.
1
2 Group-isomorphism testing
2
3 Group enumeration
3
4 Other topics
4
1
Group decompositions.
system Magma [6] handle groups of size p200 in minutes on conventional laptops (thanks to P. A. Brooksbank for most of this implementation). With a little care around problems in number theory the algorithms also apply to groups of matrices and other
customary methods to describe groups.
To come down from |G|O(log |G|) time algorithms
to (log |G|)O(1) you need more than better programming, you need mathematics. The inductive tool is
an observation about varieties V of groups and their
associated verbal V(G) and marginal V∗ (G) subgroups. A remarkable fact about these subgroups is
that if the center ζ1 (G) ≤ V∗ (G) then for every maximal direct product decomposition Q of G/ζ1 (G) and
every maximal direct product decomposition R of G,
Q refines R modulo V∗ (G). In other words, if we plan
to construct maximal decompositions by working recursively with quotients then we can almost always
afford to be greedy. Dual properties occur with V(G)
and complete account is given in [30, Section 1.2].
We say almost always because the critical base case
of this approach are groups G where there are no
varieties V, in which ζ1 (G) < V∗ (G) unless G =
V∗ (G). This behavior is only possible with nilpotent
groups of class 2, but there so many such of groups
that a completely different approach is needed.
Using the biadditive map
Groups are easy to construct by direct products and
[, ] : P/ζ1 (P ) × P/ζ1 (P ) → γ2 (P )
by related products such as central products. It is
surprisingly difficult to reverse this process. It was
an open problem to devise an algorithm that given of commutation in p-groups P of class 2 I proved:
a finite group G could detect a proper direct factor
in fewer than |G|O(log |G|) steps. This I solved by Theorem 2 Fix a finite p-group P of class 2.
introducing new functorial relationships between p1. P is directly indecomposable if, and only if,
groups and various nonassociative rings.
ζ (P ) ≤ γ (P )P p and
1
Theorem 1 There are deterministic polynomialtime algorithms that, given generators for a group of
permutations, returns a maximal direct product decomposition and a maximal central product decomposition [27, 29, 30, 32].
The total number of operations in the group
required by these algorithms is not more than
O((log |G|)8 ) and prototypes of the essential components of those algorithms in the computer algebra
2
C(P ) = {(F, G) : [uF, v] = [u, v]G = [u, vF ]}
is a local commutative ring [30].
2. P is centrally indecomposable if, and only if,
ζ1 (P ) ≤ γ2 (P )P p and
Sym(P ) = {(F, G) : [uF, v] = [u, vF ]}
is a local Hermitian Jordan ring [28, 29].
James B. Wilson
Such indecomposability criteria offer several further insights. For instance, the rings C(P ) and
Sym(P ) are completely arbitrary and so we can categorize indecomposability according to the various
types of rings that arise. This can help distinguish
groups that are not isomorphic. Also, we can compute the rings by solving systems of linear equations.
That completes the algorithms for Theorem 1. For
central products there was a further critical implication.
It has been known for a century that the automorphism group of a finite group acts transitively on
the set of maximal direct product decompositions.
Hence, direct product decompositions are essentially
equivalent. The same is not true of central product decompositions. For example, there are nonisomorphic centrally indecomposable groups whose
central products are isomorphic, such as centrally
indecomposable 2-groups R and T where the natural central product ◦ that “identifies the centers”
behaves as follows [25, Section 6]:
R◦R◦R∼
= R ◦ T.
(1)
Perplexing examples like this marked the start of a
period of inactivity for central decompositions. Using
Jordan rings as in [28] it is now easy to explain and
predict the behavior of central products. Indeed, (1)
is a variation on and old and familiar decomposition
rule for connected sums # of 2-manifolds:
RP 2 #RP 2 #RP 2 ∼
= RP 2 #T 2 .
(2)
See [28, p. 2678] and [20, p. 101]. For primes p 6= 2
more can be said.
2
2
Group-isomorphism testing
Much of my work is motivated by questions about
group isomorphism and automorphism groups. There
is no algorithm known to test if two groups of size N
are isomorphic that does not require N O(log N ) steps.
It is hoped that one day an algorithm will be devised that uses N O(1) steps. My personal objective is
riskier than that, I seek an (log N )O(1) time algorithm
similar to how Theorem 1 resolved the direct product problem. Once again p-groups of class 2 seem
to offer the greatest challenge. Indeed, Cannon and
Holt, and Babai et. al. have developed several reductions of isomorphism testing that eventually rely
on isomorphism testing of p-groups [1, 11].
A curious illustration of what to expect with pgroups came out of joint work with M. L. Lewis [19].
Using rings with involutions we prove:
Theorem 4 For every prime p > 2 and n 0:
1. There is a group H of (3×3) upper unitriangular
matrices over a field where |H| = p1.2n+O(1) and
2
H has pΩ(n ) pairwise non-isomorphic quotients
n
of size p .
2. All quotients of H of size pn have isomorphic
character tables, all nontrivial elements have order p, all non-central conjugacy classes have the
same size, and their automorphism groups are
one of just two possible options.
3. There is a deterministic polynomial-time algorithm that, given generators for groups G and
K, determines if G and K are quotients of a
Heisenberg group, and if so, determines if G and
K are isomorphic.
Theorem 3 [28] For nonabelian p-groups G whose
centers contain their commutator subgroups and So there are huge numbers of groups with no know
whose nontrivial elements have order p (so p > 2):
no differences but in (log |G|)7 steps an algorithm can
(i) The following are invariants of fully refined cen- decide isomorphism without explaining why.
Already Fitting and P. Hall [12, 14] understood
tral product decompositions of G: the size, the
the
power of characteristic subgroups in isomorphism
multiset of orders of the members, and the multests.
The difficulty has been the lack of known
tiset of orders of the centers of the members.
characteristic subgroups. E.g. Higman [16] built
(ii) The number of (Aut G)-orbits acting on the set p2n3 /2+Θ(n2 ) pairwise non-isomorphic groups that to
of fully refined central product decompositions this day have only one known proper nontrivial charcan be any positive integer.
acteristic subgroup.
James B. Wilson
To find new and computable characteristic subgroups we turn once more to nonassociative rings.
To organize these new subgroups we generalize some
ideas of Lazard [18].
A filter φ : M → 2G maps a commutative monoid
M into the subgroups of G such that
(∀s, t ∈ M )
[φs , φt ] ≤ φs+t ≤ φs ∩ φt .
(3)
Q
For every s ∈ M , φ+
s =
t∈M −0 φs+t . So for each
s ∈ M − 0, Ls = φs /φ+
s is abelian and in fact a
Z[φ0 /φ+
]-module.
Commutation
in G induces an M 0
L
graded Lie ring L(φ) = s∈M Ls , where
L L0 is 0, or
a Lie ring acting as derivations on
s6=0 Ls ( e.g.
+
the Lie ring associated to φ0 /φ0 when this group is
algebraic.) Explicitly:
3
3
Group enumeration
Many families of finite groups arise because of interest in peculiar properties or counter-examples. The
culture of counter-examples is that: “one is enough”.
In reality we sometimes need counter-examples to be
so numerous that they highlight the incredible blunder in our intuition about groups. Likewise, our intuition is better shaped when we find that speculated
“special” behavior is actually quite common.
In this spirit I have amplified the known quantities
of various families of groups. We discover that for
the most part the special properties we ascribe to
groups are in fact rather common. Two examples of
this were already seen in Theorems 4 & 5. Further
examples are indecomposable groups [32].
Theorem 6 The log of the number of pairwise nonisomorphic centrally (and directly) indecomposable fiLazard’s construction amounts to filters with M = N. nite groups is equivalent to the log of the number of
The development of several new filters is outlined in pairwise non-isomorphic finite groups.
[33]. In particular, using the radicals of the rings of Of course we should speak of actual proportions but
adjoints, centroids, and derivations, we can refine the so far the methods of counting in algebra can only
common central series of groups to characteristic se- estimate the logarithmic proportions, or correspondries of arbitrary length. Since the refinement method ingly the dimensions of varieties; see [5]. Actual carcan be made algorithmic this offers a very real im- dinalities are known only for very small families and
provement for isomorphism testing for the groups this after very difficult classifications e.g. [4, 23].
with large filters. Indeed, the number of such groups
One final example is worth mentioning arise from
is at least a positive logarithmic proportion of all p- representation theory. A Camina triple is a group G
groups as the following shows.
and subgroups M and N where every g ∈ G − N
+
+
[xφ+
s , yφt ] = [x, y]φs+t ,
s, t 6= 0.
Theorem 5 ([33]) For every prime p and n 3
2
2, there are at least p2n /729+Ω(n ) pairwise nonn
isomorphic groups of order p that admit a filter of
length at least 6 but whose classic filters all have
length 2.
An even deeper analysis of the ring theory allows for improvements of isomorphism test even when
there are no new characteristic subgroups. The idea
is to consider radical and semisimple properties of the
various nonassociative rings introduced for p-groups.
See [10]. In collaboration with P. A. Brooksbank
and E.A. O’Brien we are identifying combinatorial
techniques to construct new characteristic subgroups.
Those methods rely on techniques from [9].
has [G, g] = M . Camina pairs occur when M = N
and Camina groups when M = N = G0 . We have
seen that there are at least N Ω(log N ) pairwise nonisomorphic Camina groups, Theorem 4. Camina
triples, while more general, are still considered very
constrained examples of groups because their representation theory is highly specialized. In joint work
with T. Miyazaki, we prove the following surprising
quantity of Camina triples.
Theorem 7 The number of finite p-groups admitting a Camina triple is logarithmically dense in the
number of finite groups.
With examples like Theorem 7 and Theorem 4, we
begin to recognize that our tools for understanding
James B. Wilson
groups are quite coarse and do not yet get at most of
the distinguishing features of isomorphism types of
groups. In particular we are encouraged even more
to seek out algorithms that can identify isomorphism
since our group theory is so often inadequate.
A brief word about the proof of Theorem 7. Previous counting methods of Higman [16] and our own
in Theorems 4, 5, & 6 estimate the number of orbits
in the action of a well-known group on a well-known
set. Those methods fail when we consider groups
with unusual constraints such as the Camina triple
property.
To perform our count in Theorem 7 we have instead
changed the problem from counting to the problem
of “complexity.” Complexity is the computer science
term for expressing how hard it is to solve a problem. We show that to prove a triple (G, N, M ) has the
Camina property is an NP-complete problem, i.e. extremely difficult. From that knowledge we can apply
results about other difficult problems, such as theorem provers and Boolean satisfiabilty. By changing to
different problems we eventually encounter an equivalent problem but one for which the proportions are
known. So we are able to infer that there are many
triples with the Camina property. This technique appears to be new even though it is at the same time
obvious. We are in the middle of discover some of its
other uses.
4
Other topics
There are various branches in my research with interesting outcomes. For example, the classical subgroups of a general linear group have been known for
nearly a century and they arise all over mathematics
and science. In many situations a particular problem
occurs with more than one geometry and so the relevant group is no longer single a classical group but
the intersection of multiple classical subgroups.
Intersections of classical groups have been considered by many with progress made by Wagner, BayerFluckiger, Goldstein-Guralnick, and BrooksbankO’Brien [2,7,13,26]. In most of those works the intersection of two classical groups of the same geometry
(but in different bases) was explained but intersec-
4
tions with more geometries or with different families
of geometry was not understood.
My tools for p-groups translate naturally to intersections of classical groups and in that context I rediscovered many earlier results and made progress over
previous methods especially in the context of multiple geometries [28, Theorem 4.39, Section 7.5]. In
later joint work with P. A. Brooksbank we handle
the general case both in a theoretical and practical
manner.
Theorem 8 [8] If Φ is a set of symmetric, alternating, or Hermitian forms on a finite vector space V of
odd order, and 0 = V0 < · · · < Vt = V is a composition series of V as an ∩φ∈Φ Isom(φ)-module; then:
(i) For all i ∈ {1, . . . , t}, there is a field Fi over
which V̄i = Vi /Vi−1 becomes an Fi -vector space
and ∩φ∈Φ Isom(φ) acts on V̄i as a general classical group in its defining representation.
(ii) U = ∩φ∈Φ U (Isom(φ)) is the kernel of the action
of ∩φ∈Φ Isom(φ) on V̄1 ⊕ · · · ⊕ V̄t , and equals:
p
{z + 1 + z 2 : z ∈ End V, ∀φ ∈ Φ
∀u, v ∈ V, φ(uz, v) = −φ(u, vz),
∀i, 1 ≤ i ≤ t, Vi z ⊆ Vi−1 }.
(iii) There
are
subgroups
H1 , . . . , H m
of
∩φ∈Φ Isom(φ), such that each Hj acts faithfully
as a classical group on some (possibly several)
V̄i , and ∩φ∈Φ Isom(φ) = U o (H1 × · · · × Hm ).
There are polynomial-time algorithms to compute this
structure (now available in Magma [6]).
One of the fortunate
was to mimic the
P∞ breakthroughs
n
role that ez = n=0 zn! plays in connecting Lie algebras to Lie groups but use instead:
z+
p
1 + z2 = 1 + z − 2
∞
X
(−1)n C(n − 1)
n=1
zn
(4)
4n
where C(j) is the j-th Catalan number. As (4) only
requires division by 2, it applies to all fields of characteristic not 2 (ez does not apply so generally).
Some other directions of importance to me include
the study of semifields, which is closely related to
James B. Wilson
Camina groups, and the development of categories
derived from the properties of p-groups. Since pgroups are so vast and complicated it makes sense
to consider functorial relationships to better known
contexts. Though initially these new categories appear routine they can in fact provide contexts for new
observations.
For every distributive products ◦ : U × V → W
there is an associative ring Adj(◦) which is universal in a natural way. Since commutation in p-groups
induces such products, this abstraction is immediately applicable to p-groups. It induces a Galois
correspondence between p-groups and ∗-rings, and
defines a closure operator on p-groups. Adjoints
also determine a category which relates to chains of
double-centralizers in p-groups. The adjoint category is abelian, complete, and self-dual. However,
using Morita duality we can show this category is
not equivalent to a module category [31]. Also the
“nondegenerate-simple” objects in the adjoint category are classified and they turn out to coincide with
bilinear products without zero divisors:
(∀u ∈ U, ∀v ∈ V )
u ◦ v = 0 ⇒ u = 0 or v = 0.
5
References
[1] L. Babai, P. Codenotti, J. A. Grochow, and Y. Qiao,
Code equivalence and group isomorphism, Proc. 22nd
ACM-SIAM Symposium on Discrete Algorithms, SIAM,
Philadelphia, PA, 2011, pp. 1395–1408. MR2858409
(2012j:94191)
[2] E. Bayer-Fluckiger, Principe de Hasse faible pour les
systèmes de formes quadratiques, J. Reine Angew. Math.
378 (1987), 53–59.
[3]
, Systèmes de formes quadratiques d’algèbre
donnée, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995),
no. 10, 1291–1293 (French, with English and French summaries). MR1363566 (96m:11025)
[4] H. U. Besche, B. Eick, and E. A. O’Brien, The groups of
order at most 2000, Electron. Res. Announc. Amer. Math.
Soc. 7 (2001), 1–4. MR1826989
[5] S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of finite groups, Cambridge Tracts in
Mathematics, vol. 173, Cambridge University Press, Cambridge, 2007. MR2382539 (2009c:20041)
[6] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput.
24 (1997), no. 3-4, 235–265. Computational algebra and
number theory (London, 1993). MR1484478
[7] P. A. Brooksbank and E. A. O’Brien, On intersections of
classical groups, J. Group Theory 11 (2008), no. 4, 465–
478. MR2429348 (2009d:20115)
[8] P. A. Brooksbank and James B. Wilson, Computing isom-
etry groups of Hermitian maps, Trans. Amer. Math. Soc.
Such bilinear maps include the multiplications of
364 (2012), no. 4, 1975–1996. MR2869196
nonassociative division rings (i.e. semifields) and are
, Intersecting two classical groups, J. Algebra 353
[9]
also used to construct nontrivial elements in higher
(2012), 286–297. MR2872448
n+k
homotopy groups πn (S
). So we now see that there
[10] P. A. Brooskbank and J. B. Wilson, Groups acting on
is a critical connection between nilpotence – products
tensors (submitted). arXiv:1210.0827.
that eventually become 0 – and products that have [11] J. J. Cannon and D. F. Holt, Automorphism group compuno zero-divisors and topological constructions. See
tation and isomorphism testing in finite groups, J. Symbolic Comput. 35 (2003), no. 3, 241–267. MR1962794
[31].
(2004c:20035)
Closing
[12] H. Fitting, Beiträge zur Theorie der Gruppen endlicher
Ordnung., Jber DMV 48 (1938), 77–141.
[13] D. Goldstein and R. M. Guralnick, Alternating forms and
self-adjoint operators, J. Algebra 308 (2007), no. 1, 330–
349. MR2290925 (2008b:20050)
I have attempted to mention results accurately but
without introducing all appropriate definitions and [14] P. Hall, The construction of soluble groups, J. Reine
Angew. Math. 182 (1940), 206–214. MR0002877 (2,125j)
contexts. These should give a flavor of my work but
[15]
G.
Higman, Enumerating p-Groups, I – IV, Group Theory
it is unavoidable that by such a terse treatment some
Seminar Lectures, Department of Mathematics University
of the claims will be miss-understood or miss-stated.
of Chicago, Unknown Month 1960.
The articles are the best place for accurate detail. I [16]
, Enumerating p-groups. I. Inequalities, Proc. Lonwelcome comments on this or any other material of
don Math. Soc. (3) 10 (1960), 24–30. MR0113948 (22
#4779)
mine you find interesting or worth improving.
James B. Wilson
[17]
, Enumerating p-groups. II. Problems whose solution is PORC, Proc. London Math. Soc. (3) 10 (1960),
566–582. MR0123605 (23 #A930)
[18] M. Lazard, Sur les groupes nilpotents et les anneaux de
Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190.
MR0088496 (19,529b)
[19] M. L. Lewis and J. B. Wilson, Isomorphism in expanding families of indistinguishable groups, Groups Complex.
Cryptol. 4 (2012), no. 1, 73–110. MR2921156
[20] J. Milnor and D. Husemoller, Symmetric bilinear forms,
Springer-Verlag, New York, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. MR0506372 (58
#22129)
[21] M. F. Newman, Determination of groups of prime-power
order, Group theory (Proc. Miniconf., Australian Nat.
Univ., Canberra, 1975), Springer, Berlin, 1977, pp. 73–
84. Lecture Notes in Math., Vol. 573. MR0453862 (56
#12115)
[22] E. A. O’Brien, The p-group generation algorithm, J. Symbolic Comput. 9 (1990), no. 5-6, 677–698. Computational
group theory, Part 1. MR1075431 (91j:20050)
[23] E. A. O’Brien and M. R. Vaughan-Lee, The groups with
order p7 for odd prime p, J. Algebra 292 (2005), no. 1,
243–258. MR2166803 (2006d:20038)
[24] L. Rónyai, Computing the structure of finite algebras, J.
Symb. Comp. 9 (1990), 355-373.
[25] C. Y. Tang, On uniqueness of central decompositions of
groups, Pacific J. Math. 33 (1970), 749–761. MR0265472
(42 #381)
[26] A. Wagner, On the classification of the classical groups,
Math. Z. 97 (1967), 66–76. MR0209366 (35 #264)
[27] J. B. Wilson, Group decompositions, Jordan algebras, and
algorithms for p-groups, ProQuest LLC, Ann Arbor, MI,
2008. Thesis (Ph.D.)–University of Oregon. MR2712085
[28]
, Decomposing p-groups via Jordan algebras, J.
Algebra 322 (2009), no. 8, 2642–2679.
MR2559855
(2010i:20016)
[29]
, Finding central decompositions of p-groups, J.
Group Theory 12 (2009), no. 6, 813–830. MR2582050
(2011a:20044)
[30]
, Existence, algorithms, and asymptotics of direct
product decompositions, I, Groups Complex. Cryptol. 4
(2012), no. 1, 33–72. MR2921155
[31]
, Division, adjoints, and dualities of bilinear
maps (in press), Communications in Algebra 41 (2013).
arXiv:1007.4329.
[32]
, Existence, algorithms, and asymptotics of direct
product decompositions, II, Groups Complex. Cryptol. (to
appear, arXiv:1005.0548).
[33]
,
Longer
arXiv:1303.3157.
nilpotent
series
(submitted).
6