Computational Complexity in Finite Groups
Làszló Babai *
Department of Computer Science, University of Chicago, Chicago, IL 60637, USA
and Department of Algebra, Eötvös University, Budapest, Hungary H-1088
We survey recent results on the asymptotic complexity of some of the fundamental
computational tasks in finite groups in a variety of computational models. A
striking recent feature is that techniques motivated by the problems of the
more abstract models (nondeterminism, extreme parallelization) have turned out
to provide powerful tools in the design of surprisingly efficient algorithms on
realistic models (e.g. a nearly linear time membership test for permutation groups
with a small base).
The techniques involve a combination of elementary combinatorial results on
finite groups, some classical elementary group theory, and the extensive use of
certain consequences of the classification of finite simple groups (CFSG).
Most of the recent work surveyed is due to E. M. Luks, G. Cooperman,
L. Finkelstein, A. Seress, E. Szemerédi, and the author.
1. Group Models and Measures of Complexity
Rubik's Cube illustrates some of the basic problems of computational group
theory. We may want to decide whether or not a particular configuration is
feasible (accessible without pulling the cube apart) ; determine the total number of
feasible configurations; or construct "typical" configurations. In group theoretical
terms, we are given a group G by a list S of generators (the "legal moves"), and
we wish to determine whether or not a particular element of a larger group
belongs to G (membership testing)', determine the order of G; generate uniformly
distributed random members of G. The gourmet will ask more sophisticated
questions such as deciding solvability, nilpotence, constructing normal closures,
the center, composition factors, Sylow subgroups, etc.
The cost of answering these questions depends on two factors : the way group
operations are performed, and the measure of cost.
For greatest generality, we consider black box groups, a model where no
restriction is made on the way group operations are performed. In this model,
elements of an unknown group B (the "group in the box") are encoded by
* Research supported in part by NSF Grant CCR-8710078 and Hungarian National
Foundation for Scientific Research Grant 1812.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
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Laszló Babai
binary strings of uniform length n. (In particular, \B\ < 2n.) Group operations
are performed by the "black box" at unit cost. A black box group is a subgroup
of B given by a list of generators. (Generators of B, or a recognition method of
strings in B, are not assumed to be known.)
Two implementations of the "black box" are of particular interest: permutation
groups (subgroups of the symmetric group B = Sym(ß)), and matrix groups over
finite fields (subgroups of the linear group B = GL(d,q)).
The models of computation to be studied include deterministic as well as
Monte Carlo (randomized) computation, both sequential and parallel (several
processors). Timing estimates refer to the logarithmic cost RAM model [AHU].
A Monte Carlo algorithm uses randomization, hence its outcome may be in
error. However, on any input, the probability of error is required to be < 1/4. By
repeating the algorithm m times and taking majority vote, the chance of error is
reduced to < e~m^ (Chernoff's bound).
The cost (in terms of a specific resource such as time, space, number of
processors, length of proofs) is measured as a function of the length of the input,
i.e. the number of input bits. An algorithm is said to have cost 0(f(n)) (or cost
0~(f(n))) if for n>no and on all inputs of length n, the cost is at most cf(n) (at
most /(n)(logn) c , resp.) (no, c will denote various constants throughout).
A function f(n) is polynomially bounded (or "short", "small") if f(n) '< nc for
some c and all n > no. A polylog bound means f(n) < (logn)c. When used as
technical terms, "short", "small" will be Italicized.
NC ("Nick's Class") denotes (somewhat informally) the class of functions
computable in polylog time, using a small number of parallel processors.
NP ("nondeterministic polynomial time") stands for the class of decision
problems where the "yes" answers have short proofs. More precisely, a predicate
^4(x) belongs to NP if there exists a polynomial time computable predicate
B(x, w) such that for every input string x, ^4(x) <-» (3pw)B(x, w), where 3pw refers
to short strings w. The string w is called a witness of the statement A(x). The
negations of NP -predicates form the class coNP. (Cf. [GJ].)
We shall also consider the class AM ("Arthur-Merlin"), a randomized extension of NP, defined as follows: the predicate ^4(x) belongs to AM if there exists
an NP -predicate B(x,r) such that for every input string x, ^4(x) is equivalent to
B(x,r) for most short strings r. (The definition of "most" is flexible; asking more
than 51% will define the same complexity class as asking, say, a 1 — 2~~n fraction, where n is the length of x.) Informally, A(x) has short "interactive proofs"
in the sense that if the all-knowing but untrusted Merlin is able to present a
short "witness" in response to a random question r of polynomial time bounded
Arthur, this should convince skeptical Arthur by way of overwhelming statistical
evidence that Merlin's claim A(x) is true. (Cf. [Ba2, BM, GMR, Go].)
2. General Methods: Black Box Groups
In this section we demonstrate the somewhat unexpected fact that nontrivial
computational tasks, such as constructing random elements and deciding solvability, can be accomplished in Monte Carlo polynomial time in the extremely
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general model of black box groups. Here and throughout the paper, G will denote
a finite group and S a set of generators of G.
2.1 Two Combinatorial Lemmas
We begin with two elementary results which will play a key role. The first one
concerns the number of group operations required in order to construct an
element from a given set of generators.
A straight line program from S ç G t o g G G i s a sequence u\,...,um of
elements of the group G such that each u\ either belongs to S or is obtained from
one or two previous elements by a group operation; and g = um. The straight
line cost cost(g\S) is the smallest m such that such a straight line program exists.
Lemma 1 (Reachability Lemma [BSz]). Given any set S of generators of a group
G and any g e G, we have cosl(g|S') < (1 + log \G\)2.
(All log's in this paper are to the base 2.) A subproduct of the elements
/ii,...,/i/c e G is a product of the form ltf -'hekk, where e\ e {0,1}. The kdimensional cube C = C(h\,...,hk) is the set of all the 2k subproducts of the h\.
The cube C is nondegenerate if \C\ = 2k. The basic structure established in the
proof of Lemma 1 is a chain of nondegenerate cubes: we prove the existence of
a sequence of elements h\,...,ht which generate a nondegenerate cube C such
that G = C~XC\ and for every /, cost(/?/|/ii,...9fy_i,S) < 2/— 1. The h\ are found
inductively; we can continue as long as C~XC ^ G: the element outside C~~lC of
lowest straight line cost will do. We shall refer to this procedure as doubling the
cube. Clearly, we must stop at some t < log \G\.
The proof just sketched is non-constructive; it does not tell how to find an
element that will double the current cube. The following lemma provides the key
to an efficient Monte Carlo procedure.
A graph is a pair X = (V,E) where V is the set of vertices, and E is a
set of unordered pairs of vertices, called edges. Two vertices v, v' are adjacent
if {v, v'} E E. An isomorphism of two graphs is a bijection of the vertex sets
preserving adjacency. The group of self-isomorphisms of X is the automorphism
group A\xt(X). We say that X is vertex-transitive if Aut(X) is a transitive subgroup
of Sym(F). The number of vertices adjacent to y G F is the degree of v. X is
locally finite if its vertices have finite degrees. The boundary dW of a subset
W Ç V consists of all vertices in V \ W adjacent to some vertex in W. X is
connected if dW ^ 0 for any nonempty proper subset W. A walk of length £ in
X is a chain of *f + 1 vertices, each adjacent to its predecessor. The distance of
v,v' e V is the length of the shortest walk between them. Let X\v) denote the
set of vertices at distance < t from v.
Lemma 2 (Local Expansion Lemma [Ba4,5]). Let X = (V,E) be a locally finite
connected vertex-transitive graph and veV.IfW^
X\v) and \W\ < \V\/2 then
\dW\>\W\/(At).
This lemma has the interesting consequence that random walks on a vertextransitive graph "don't get stuck" in a corner.
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Laszló Babai
Theorem 3 [Ba5]. Let X be a connected vertex-transitive graph of finite degree d.
Assume that \XM(v)\ < \V\/2. Let % be a random integer chosen uniformly from
{t,t+ !,...,£} where £ = Ct2dlog\G\ (C = 500). Then with probability > 1/16, a
random walk of length %, starting at v, will end outside Xl(v).
Among the ingredients of the proof is a Cheeger-type [Ch] eigenvalue estimate
for graphs, derived from the local expansion property, following the Unes of [Alo].
The graphs this theorem will be applied to are Cayley graphs. The vertex set
of the Cayley graph X(G, S) is G; and g e Gis adjacent to gh for he SU S~{.
2.2 Membership and Random Generation
We begin with a nondeterministic result.
Theorem 4 [BSz]. Membership in black box groups belongs to NP.
Indeed, a short straight line program qualifies as a witness of membership.
There is little hope for making this proof constructive, even in the very special
case when the "group in the box" is the multiplicative group of GF(q). Indeed
in this case, finding a straight line program to generate g from S = {h} is
equivalent to solving the equation hx = g. This is the discrete logarithm problem,
not believed to be solvable in polynomial time. (Known algorithms require time
Gxp(c^/qlogq); whereas polynomial time would mean polylog(g) steps.)
Yet, part of the proof can be turned into an efficient Monte Carlo algorithm.
Theorems [Ba5]. Nearly uniformly distributed random elements of a black box
group can be constructed in Monte Carlo polynomial time.
Nearly uniform distribution means each element has probability (1 ±ß)/|G|
to be selected; and the reliability of the algorithm is > 1 — ö, where e, ö are
input parameters, and the number of operations is polynomially bounded in
k = log |G|+log(l/e)+log(l/<5). The idea is to construct a set Sf of generators such
that the diameter of the Cayley graph X(G, Sf) is small. Once this is accomplished,
short random walks are known to produce nearly uniformly distributed elements
[Aid, Alo].
To reduce the diameter, we should like to adapt the "doubling the cube" trick.
The difficulty is, how to obtain the next hu which must be outside the set C~lC,
where C is the current cube. The solution is a short random walk of random
length over the Cayley graph. By Theorem 3, such a walk has a fixed positive
chance of reaching a desired element.
2.3 Nonmembership, Order
These two problems are known to belong to the class AM [Ba4]. Indeed, to prove
this was the original motivation behind inventing the Local Expansion Lemma.
For black box groups, the nonmembership and order verification problems are
provably not in NP [BSz]. (To be precise, here we are talking about a relativized
version of NP: computations refer to the black box, an "oracle" [GJ].)
Computational Complexity in Finite Groups
1483
We conjecture, however, that for matrix groups over finite fields, these problems belong to NP. This will follow from the conjecture below. The length of a
presentation of a group (in terms of generators and relations) is the number of
bits required to write down the presentation. E. g., the presentation (a\aN = 1)
of the cyclic group has length log N + 0(1) (we write exponents in binary).
Short Presentation Conjecture, (i) Every finite simple group G has a presentation
R of length polylog(|G|). (ii) If G is of Lie type over GF(q) then such R is
computable from the standard name of G in time polylog(|G|), assuming GF(q)
and a primitive root in it are explicitly given.
One can prove, using Lemma 1, that part (i) of the Conjecture, if true,
automatically extends to all finite groups [BKLP], The conjecture itself has been
verified for all G except those of rank one twisted Lie type [BKLP], cf. [Ka4].
Note that for a Lie-type simple group of rank d over GF(q), the Conjecture
requires presentations of length < (dlogq)c; whereas the Steinberg presentations
[St, Car] require an exponentially greater number, about d2q generators.
Verification of the order is a central problem. If it belongs to NP (as we
expect for matrix groups), this brings a number of other verification problems
into NP, including composition factors, homomorphisms, isomorphisms, kernels,
minimal normal subgroups. On the other hand, problems known to be in AM
(for black box groups in general) but not expected to belong to NP (even for
permutation groups) include verification of the intersection of two subgroups, the
centralizer of an element, non-conjugacy of two elements [Ba4].
2.4 Random Subproducts
Algorithms often depend on access to random elements of G (e.g. [CFS, NP]).
Theorem 5 constructs such elements in reasonable polynomial time, but not
efficiently enough for some applications. Random subproducts, however, often
emulate truly random elements very efficiently.
Let S = {gi,...,gs}- A random subproduct is a subproduct cp = g\x •••g|%
where the ej are independent uniform (0, Invariables (coin flips).
Lemma 6 [BLS2]. Let H < G be a proper subgroup. Then Prob(<p ^ H) > 1/2.
Let L be the maximum length of subgroup chains in G. (L <> log |G|.) Lemma
6 implies that the probability that 2L(1 + a) random subproducts do not generate
G is less than exp(—a2L/(l + a)). A refined argument yields:
Lemma 7 [BCFLS]. / / G is given by a list of s generators, then a Monte Carlo
procedure, using 0(s\ogL) group operations, produces (with large butfixedprobability) a set of 0(L) generators for G.
This keeps the number of generators down when constructing subgroups.
Some additional combinatorics yields a particularly efficient normal closure algorithm. Recall that the notation 0~(f(n)) refers to an upper bound f(n)(iogn)°^\
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Laszló Babai
Theorem 8 [BCFLS]. Let H < G be black box groups, each given by a list ofO(L)
generators. Then the normal closure of H in G can be constructed (in the form of
0(L) generators) in Monte Carlo 0~(L2) operations.
As an immediate application, we obtain polynomial time Monte Carlo algorithms to decide solvability and nilpotence of G (both require 0~(L3) operations).
We should stress that the results of this section apply to all black box groups,
including their implementations as matrix groups or permutation groups. In spite
of their generality, the results are strong enough to yield asymptotic savings even
in the well-studied area of permutation groups (see Sect. 4.2).
3. Permutation Groups: Survey of Complexity Status
We consider groups G < Sym(ß) where |Q| = n. The stabilizer of x G Q is the
subgroup Gx = {g G G : xg = x}. The pointwise set stabilizer of A e Q is the
subgroup G A = f]XeA Gx. We call A a base if G A = {1}.
Let G = G® > G® > ... > G<m> =t {1} be a chain-of subgroups. A strong
generating set (SGS) w.r. to this chain is a set S ç G such that G(i) = (Sn G®)
for every i. A transversal system (TS) is a family {Tt : 1 < i < m], where Ti
is a (right) transversal (set of coset representatives) of G® in G^~l\ A partial
transversal system is a family of partial transversals T{ ^ Tt.
The stabilizer chain w.r. to a given ordered base zl = {xi,...,x m } is defined as
G® = GXlv..jX(. The concept of an SGS w.r. to an ordered base was introduced by
C. G Sims in the early 60's as a central tool in computational group theory ([Siml,
2]). Given an SGS, a TS is readily constructed, solving the membership and order
problems and the construction of truly uniformly distributed random elements. A
slight modification yields normal closures, clearing the way for more advanced
applications. The central problem of constructing an SGS was efficiently solved
by Sims [Siml, 2],
The asymptotic complexity of these algorithms was not analyzed until 1980
when the complexity of many of these algorithms was recognized to be polynomial
time in [FHL]. In particular, an 0(n6 + sn2) variant of Sims's SGS algorithm
was constructed, where s is the number of input generators. As a consequence,
membership, order, normal closures, solvability were shown to be computable
in polynomial time [FHL]. E. M. Luks has subsequently added an array of
elegant polynomial time algorithms which, for the first time, required deeper
group theoretic analysis. The list includes the center, a composition chain [Lu2],
and subcases of the coset intersection problem, i.e. determining G n H h where
G, H < Sym(ß), h e Sym(ß). The subcases solved in polynomial time in Luks's
seminal paper [Lui] include the case when G is solvable, or more generally, the
nonabelian composition factors of G are restricted to the set ^(c) consisting of
the alternating groups of degree < c, the groups of Lie type of rank < c, and
the sporadic groups. The algorithm uses classical divide and conquer algorithmic
techniques, splitting the domain into orbits and then into domains of imprimitivity
([Wi]). When G is primitive, some of the algorithms use exhaustive search. In
such cases, the polynomial time claim depends on the following result.
Computational Complexity in Finite Groups
1485
Theorem 9 [BCP], If G < Sym(ß) is primitive and G e $(c) then G is small.
(Small means \G\ < if' for some constant c', depending on c.) For primitive
solvable groups, the precise bound is \G\ < 24_1/3/7c where c = l+log9(48-241/3) =
3.24399... [Pâ, Wo].
Sylow subgroups and Sylow normalizers were added to the polynomial time
library by Kantor [Kai, 2, 3]. For a long list of additional results see [KL].
Another important observation of [Lui] was that a number of problems,
including coset intersection, setwise stabilizer of a subset, centralizer of an element,
and centralizer of a subgroup, are equivalent (polynomial time reducible to one
another), and the graph isomorphism problem (to decide whether or not two
given graphs are isomorphic) is reducible to each. In particular, as long as graph
isomorphism is not solved in polynomial time (the best current algorithm requires
exp(0~(s/ri)) for graphs on n vertices, cf. [BL]), coset intersection, etc., are not
expected to be efficiently solvable. On the other hand, the decision versions of
these problems ("is GnHh ^ 0?") are not NP -complete, unless the so-called
polynomial time hierarchy of complexity classes collapses [BM, GMW]. (The
conjecture that the polynomial time hierarchy does not collapse is a stronger
version of the famous NP =fc coNP conjecture [Sto].)
Some related problems are ATP-complete; the nicest is A. Lubiw's result: the
predicate "G has a fixed-point-free element" is NP-complete, even for elementary abelian 2-groups [Lub]. An even harder problem is to determine minimum
generating sequences; the length of the shortest word in S representing g e G is
PSPACE-completQ pel]. (For related problems, see [BHKLS].)
On the other end of the spectrum, some of the basic problems were shown
to admit ultra-fast parallel algorithms. Most notably, membership, order, and even
a composition chain are computable in NC [BLS1]. A striking feature of the
algorithm is that even for the rudimentary tasks of membership testing, we are
forced to determine the composition factors first, using several facts of asymptotic
group theory currently derivable only via the classification of finite simple groups
(CFSG) (cf. Sect. 5.). - Coset intersection is not known to be in NC; if it is in
NC, then so is the isomorphism of graphs of degree 3 [LM].
4. Efficient Construction of Strong Generators
The 0(n6 + sn2) analysis of the SGS algorithm of [FHL] was soon replaced by
0(n5 + sn2) [Kn, Je2]. Knuth's is the closest to Sims's original approach and is
quite efficient in practice, but there exist large collections of examples where its
typical behavior is as bad as the n5 worst case bound (while s = n — 1) [Kn].
The n5 bottleneck was broken, using machinery developed for the NC result,
in [BLS2] (0~(n4 + sn2)). Further improvements yield 0~(sn3) [BLS3], the best
deterministic bound to date. These results heavily depend on the CFSG.
4.1 A Fast and Elementary Monte Carlo SGS Algorithm
We outline a new 0~(/73 + sn) time Monte Carlo SGS algorithm with a perfectly
elementary analysis [BCFLS].
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Laszló Babai
Let H < G be a subgroup of index k; and T = {t\,..., tk} a right transversal.
For g e G, let g = t,- where iîg = H^. Schreier's lemma asserts that the sk
elements Uhjtihj ("Schreier generators") generate H, where S = {h\,...,hs}
[Ha]. Noting that a transversal for Gx in G is easily constructed, a natural
approach to constructing an SGS would be to consider the Schreir generators for
Gx and repeat. The difficulty is that the number of generators grows rapidly.
Random subproducts are the new tool. (When using the results of Sect. 2.4,
the following bound comes handy: Every subgroup chain in Sn has length < 2n
[Ba3, CST].) Lemma 7 alone saves nearly an order of magnitude over [Kn, Je2]:
we reduce the number of generators of G to 0(n), then construct a transversal
and the Schreier generators of Gx; repeat. The cost of this naive approach is
0~(n4 + sn). When G > An, we have a particularly speedy variant, based on the
following consequence of Lemma 6.
Proposition 10 [BLS2]. If G = (S) < Sn and Sf is a set ofclogn random subproducts of S then with large probability, the orbits of (Sf) and G agree.
Applying this to the action of G > An on the set of ordered 6-tuples we find
that O(logn) random subproducts are likely to generate all of G. So the previous
argument, using only 0(logn) random subproducts in each round, constructs an
SGS for G > An in time 0~(n3). This process works also when G induces Ak
or Sk on some orbit A s Q (\A\ = k). However, now we don't get generators
of G A. (The procedure preserves the action of G on A only.) Instead, we use a
set of 0(k) defining relations of Ak or Sk to construct normal generators of G A',
and then use our normal closure algorithm (Theorem 8) to obtain GA. Another
ingredient of the 0~(n3 +sn) algorithm is computation of Gx in 0~(snk), where k
is the length of the orbit of x (apply Lemma 7 to the Schreier generators). This
bound is exploited through a "smallest orbit first" strategy. By adding action on
maximal blocks to Q we ensure that the next x to be stabilized is from an orbit
with primitive action. The timing depends on a combinatorial observation:
Lemma 11 [BCFLS]. IfG<Sn is primitive and Gx has a nontrivial orbit of length
k < n/2 then every subgroup H ^ {1} of Gx has a nontrivial orbit of length < k.
4.2 Small Base Groups in Nearly Linear Time
Groups with a small base are of particular importance; e.g. linear groups, treated
as permutation groups on a vector space, always have a base of size < log n. Let
us say that a family of groups has small bases if they have base size polylog(n).
The SGS methods of [Sil,2], [Kn], [Je2], [BCFLS] require > n2 time for such
groups. Combinatorial techniques based on Lemmas 1 and 2 have recently led to
a Monte Carlo SGS algorithm in time 0~(n) for small base groups [BCFS].
The basic ideas are (i) a very efficient implementation of Sims's "Schreier
vector" data structure to store transversals, based on the "doubling the cube"trick
(Sect. 2.1) ; and (ii) the use of Lemma 2 to rapidly locate elements not yet reached
by the current partial transversal system. A key new feature of these methods
Computational Complexity in Finite Groups
1487
is that rather than operating with the coarse subgroup structure, we are able to
handle chains of certain subsets, such as cubes and their generalizations.
5. CFSG vs. Elementary
We mention some of the consequences of the simple groups classification (CFSG)
used in the analysis of the algorithms quoted. Schreier's conjecture that the outer
automorphism groups of simple groups are solvable, is used in Luks's composition
chain algorithm [Lu2] and the algorithms building on it [BLS1, 2, 3, Kal, 2, 3].
In Sect, 4.1 we used that the degree of transitivity of G < S„ is t < 6 (unless
G > An) [CKS], At the cost of some extra log factors (swallowed by the 0 "
notation) this can be replaced by the 19th century bound t = o(log2n) [Jo]. Using
the CFSG, Cameron has shown that if G is a primitive group of order > n2logn
then n = (J) and G is a subgroup of SuwrSm with socle A!$ acting on the ordered
m-tuples of/-subsets of a /c-set [Cam]. This result helps reduce the case of "large"
primitive groups to Sn ; the remaining primitive groups have small bases. This is
indispensable for [BLS1, 2, 3], even if all we need is to test membership ! Kantor
uses detailed knowledge of the CFSG even just to find an element of order p.
Other elementary estimates that may help avoid CFSG references (at a cost
of some extra log's) include the bound \G\ < exp(4y^?log2n) for G primitive
but not doubly transitive [Bal] and \G\ < nclog " for G ^ An doubly transitive
[Py]. Bochert's 1892 estimate [Bo] that a doubly transitive group G ^ An has
minimal degree > n/A (cf. [Wi]) is used in [BCFS]. Combinatorial proofs may
directly suggest efficient algorithms. A case in point is the algorithm derived from
a simple proof of Jordan's o(log2n) bound on the degree of transitivity [BS],
allowing ultra-parallelized (NC) management of Sn [BLS1].
Conclusion. During the past decade, the asymptotic complexity of computation
in finite groups has been analyzed in a variety of models of computation. New
combinatorial and algebraic tools have been developed. Structural insights gained
from the study of models ranging from the unrealistic (extreme parallelization :
NC) to the absurd (nondeterminism : NP, AM) have contributed to the design of
new efficient algorithms with a reasonable expectation of competitive implementations.
Acknowledgment. I feel privileged to have had an exciting ongoing collaboration with Gene
Luks, now for over a decade. I have also greatly benefited from joint work with Gene
Cooperman, Larry Finkelstein, and Akos Seress.
References
[AHU]
[Aid]
[Alo]
A. Alio, J. Hopcroft, J. Ullman: The design and analysis of computer algorithms. Addison-Wesley 1974
D. Aldous : On the Markov chain simulation method for uniform combinatorial
distributions etc. Probab. Eng. Info. Sci. 1 (1987) 33-46
N. Alon: Eigenvalues and expanders. Combinatorica 6 (1986) 83-96
1488
[Bal]
Laszló Babai
L. Babai: On the order of uniprimitive permutation groups. Ann. Math. 113
(1981) 553-568
[Ba2]
L. Babai: Trading group theory for randomness. Proc. 17th ACM STOC,
Providence RI, 1985, pp. 421-429
[Ba3]
L. Babai: On the length of chains of subgroups in the symmetric group. Comm.
Algebra 14 (1986) 1729-1736
[Ba4]
L. Babai: Bounded-round interactive proofs in finite groups. SIAM J. Discr.
Math, (to appear)
[Ba5]
L. Babai: Local expansion of vertex-transitive graphs and random generation
in finite groups. Proc. 23rd ACM STOC 1991, pp'. 164-174
[BCFLS] L. Babai, G. Cooperman, L. Finkelstein, E.M. Luks, ÂSeress: Fast Monte
Carlo algorithms for permutation groups. 23rd ACM STOC 1991, pp. 90-100
[BCFS] L. Babai, G. Cooperman, L. Finkelstein, A. Seress: Permutation groups with
small base in almost linear time. In: Proc. ISSAC '91, Bonn (to appear)
[BCP]
L. Babai, P.J. Cameron, P.P. Pâlfy: On the orders of primitive groups with
restricted nonabelian composition factors. J. Algebra 79 (1982) 161-168
[BHKLS] L. Babai, G. Hetyei, W. M. Kantor, A. Lubotzky, Â. Seress: On the diameter
of finite groups. Proc. 31st IEEE FOCS 1990, pp. 857-865
[BKLP] L. Babai, W. M. Kantor, E. M. Luks, P. P. Pâlfy: Short presentations for finite
groups. In preparation
[BL]
L. Babai, E.M. Luks: Canonical labeling of graphs. Proc. 15th ACM STOC
1983, pp. 171-183
[BLS1]
L. Babai, E. Luks, Â. Seress: Permutation Groups in NC. Proc. 19th ACM
STOC 1987, pp. 409-420
[BLS2]
L. Babai, E. Luks, A. Seress : Fast management of permutation groups. Proc.
28th IEEE FOCS 1988, pp. 272-282
[BLS3]
L. Babai, E. Luks, A. Seress : Fast deterministic management of permutation
groups. In preparation
[BM]
L. Babai, S. Moran: Arthur-Merlin games: a randomized proof system, and a
hierarchy of complexity classes. J. Comp. Sys. Sci. 36 (1988) 254-276
[Bo]
A. Bochert: Ueber die Classe der Transitiven Substitutionsgruppen. Math. Ann.
40 (1892) 176-193
[BS]
L. Babai, A. Seress : On the degree of transitivity of permutation groups : A
short proof. J. Comb. Theory A 45 (1987) 310-315
[BSz]
L. Babai, E. Szemerédi: On the complexity of matrix group problems I. Proc.
25th IEEE FOCS, Palm Beach, FL, 1984, pp. 229-240
[Cam]
P.J. Cameron: Finite permutation groups andfinitesimple groups. Bull. London
Math Soc. 13 (1981) 1-22
[Car]
R. Carter: Simple groups of Lie type. Wiley 1972, 1989
[Ch]
J. Cheeger: A lower bound for the smallest eigenvalue of the Laplacian. In:
Problems in Analysis. Princeton Univ. Press 1970, pp. 195-199
[CFS]
G. Cooperman, L. Finkelstein, N. Sarawagi: A random base change algorithm
for permutation groups. In: Proc. ISSAC 1990, Tokyo, ACM Press and AddisonWesley 1990, pp. 161-168
[CKS]
C.W. Curtis, W.M. Kantor, G. Seitz: The 2-transitive permutation representations of the finite Chevalley groups. Trans. A.M.S. 218 (1976) 1-57
[CST]
P.J. Cameron, R. Solomon, A. Turull: Chains of subgroups in symmetric
groups. J. Algebra 127 (1989) 340-352
[FHL]
M. L. Fürst, J. Hopcroft, E. M. Luks : Polynomial-time algorithms for permutation groups. 21st IEEE FOCS 1980, pp. 36-41
[GJ]
M. Garey, D.S. Johnson: Computers and Intractability: A guide to the theory
of NP-completeness. Freeman, New York 1979
Computational Complexity in Finite Groups
[GMR]
[GMW]
[Go]
[Ha]
[Jel]
[Je2]
[Jo]
[Kai]
[Ka2]
[Ka3]
[Ka4]
[KL]
[Kn]
[Lub]
[Lui]
[Lu2]
[LM]
[NP]
[Pa]
[Py]
[Siml]
[Sim2]
[St]
[Sto]
[Wi]
[Wo]
1489
S. Goldwasser, S, Micali, C. Rackoff: The knowledge complexity of interactive
proofs. SIAM J. Comp. 18 (1989) 186-208
O. Goldreich, S. Micali, A. Wigderson: Proofs that yield nothing but their
validity and a methodology of cryptographic protocol design. Proc. 27th IEEE
FOCS 1986, pp. 174-187
S. Goldwasser: Interactive proofs and applications. These proceedings, pp.
1521-1535
M. Hall, Jr.: The theory of groups. Macmillan, New York 1959
M. R. Jerrum: The complexity offindingminimum length generator sequences.
Theor. Comp. Sci. 36 (1985) 265-289
M.R. Jerrum: A compact representation for permutation groups. J. Algorithms
7 (1986) 60-78
C. Jordan: Nouvelles recherches sur la limite de transitivité des groupes non
alternés. Bull. Soc. Math. France 1 (1873) 35-60
W.M. Kantor: Polynomial-time algorithms for finding elements of prime order
and Sylow subgroups. J. Algorithms 6 (1985) 478-514
W.M. Kantor: Sylow's theorem in polynomial time. J. Comp. Sys. Sci. 30 (1985)
359-394
W.M. Kantor: Finding Sylow normalizers in polynomial time. J. Algorithms 11
(1990) 523-563
W.M. Kantor: Some topics in asymptotic group theory. Proc. Durham Group
Theory Conf., 1990. (To appear)
W.M. Kantor, E. M. Luks: Computing in quotient groups. In: Proc. 22nd ACM
STOC, Baltimore, 1990, pp. 524-534
D.E. Knuth: Efficient representation of perm groups. Combinatorica 11 (1991)
57-68 (preliminary version circulated since 1981)
A. Lubiw: Some NP -complete problems similar to graph isomorphism. SIAM
J. Comp. 10(1981) 11-21
E.M. Luks: Isomorphism of graphs of bounded valence can be tested in
polynomial time. J. Comp. Sys. Sci. 25 (1982) 42-65
E.M. Luks: Computing the composition factors of a permutation group in
polynomial time. Combinatorica 7 (1987) 87-99
E.M. Luks, P. McKenzie: Fast parallel computation with permutation groups.
Proc. 27th IEEE FOCS 1985, pp. 505-514
P.M. Neumann, Cheryl E. Praeger: A recognition algorithm for the special
linear groups. Manuscript, 1990
P.P. Pâlfy: A polynomial bound on the orders of primitive solvable groups. J.
Algebra 77 (1982) 127-137
L. Pyber: The orders of doubly transitive groups: elementary estimates. (To
appear)
C.C. Sims: Computation with permutation groups. In: Proc. Second Symp.
Symb. Algeb. Manipulation. ACM, New York 1971, pp. 23-28
C.C Sims: Some group theoretic algorithms. (Lecture Notes in Mathematics
697.) Springer, Berlin Heidelberg New York 1978, pp. 108-124
R. Steinberg: Generators for simple groups. Can. J. Math. 14 (1962) 277-283
L. Stockmeyer: The polynomial-time hierarchy. Theoret. Comp. Sci. 3 (1976)
1-22
H. Wielandt: Finite permutation groups. Acad. Press, New York 1964
TR. Wolf: Solvable and nilpotent subgroups of GL(n,qm). Can. J. Math. 34
(1982) 1097-1111