## Math.uni-bielefeld.de

ON TRIPLE FACTORISATIONS OF FINITE GROUPS
School of Mathematics and Statistics, The University of Western Australia, Crawley WA 6009, AUSTRALIA
The second criterion characterises triple factorisations of G in terms of a subset of ΩB having
Let G, Ω, ∆ and be as in Notation 1. Then the Embedding Theorem for imprimitive permutation
restricted movement. For a finite subset Γ of Ω, the movement of Γ under the action of a group G
This poster∗ introduces and develops a general framework for studying triple factorisations of the
groups allows us to identify G as a subgroup G0 G1 with A as a subgroup of ˆ
on Ω is defined by move(Γ) := maxg∈G |Γg \Γ|. If move(Γ) < |Γ|, then Γ is said to have restricted
form G = ABA of finite groups G, with A and B subgroups of G. We call such a factorisation
our main result requires an extension of the Embedding Theorem (see Theorem 2.1 in [1]):
movement. In other words, Γg ∩ Γ = ∅ for all g ∈ G.

nondegenerate if G = AB. Consideration of the action of G by right multiplication on the right
Theorem 3 (The Extended Embedding Theorem). Let G ≤ Sym(Ω) be transitive and let Σ, ∆,
cosets of B leads to a nontrivial upper bound for |G| by applying results about subsets of restricted
A be as in (a)-(c) above. Suppose also that B ≤ G, and let B1 = BΣ,
Let A and B be proper subgroups of a group G. Consider the right coset action of G on
movement. For A < C < G and B < D < G the factorisation G = CDC may be degenerate
0 = B∆. Then there is a permutation embedding (ϕ, ψ) of (G, Ω) into (G
B, and set β := B ∈ ΩB. Then, T = (G, A, B) is a triple factorization if and only if the
even if G = ABA is nondegenerate. Similarly forming quotients may lead to degenerate triple
-orbit βA has restricted movement (see Proposition 3.2 in [1]).

1 is transitive then ϕ may be chosen such that ϕ(B) ≤ (B0 B1).

factorisations. A rationale is given for reducing the study of nondegenerate triple factorisationsto those in which G acts faithfully and primitively on the cosets of A. This involves study of a
Application of restricted movement criterion and the classification result in [16] yields a
wreath product construction for triple factorisations.

non-trivial improvement to upper bound of |G| (see [1, Theorem 1.1]):
∗ This poster is part of PhD project of the first author. Both authors are grateful to John Bamberg for helpful advice. The first author is grateful for support of an
Theorem 4. Let T = (G, A, B) be a nondegenerate triple factorisation with core
International Postgraduate Research Scholarship (IPRS), The University of Western Australia (UWA) and Summer School: FSAGRGA. This project forms part of
Australian Research Council Discovery Grant DP0770915. The second author is supported by Australian Research Federation Fellowship FF0776186.

suppose that A < H < G, and G ≤ G0 G1 with G0 = H/core
above. Then there are triple factorisations T0 for G0, T1 for G1 and T0 T1 for G0 G1 such thatT0 is a quotient of T |H, T1 is a quotient of a lift of T , T0 T1 is nondegenerate, and either
In this note we initiate a general theory of triple factorisations T = (G, A, B) of the form
(b) the triple factorisations T0 and T0 T1 are both nondegenerate, and the restriction of T0 T1 to
G = ABA for finite groups G and subgroups A, B. A special case of such factorisations was
It is tempting to replace proper subgroups A, B in a nondegenerate triple factorisation T =
G is a nondegenerate lift (G, A, D) of T .

introduced by Daniel Gorenstein [9] in 1959, namely independent ABA-groups in which every
(G, A, B) by ‘maximal overgroups’, that is, by subgroups C, D where A ≤ C, B ≤ D, and C, D
element not in A can be written uniquely as abc with a, c ∈ A and b ∈ B (see [9], also [10, 11]).

Rationale for primitive triple factorisations
are maximal subgroups of G. This guarantees that the lift (G, C, D) is a triple factorisation, but
Triple factorisations also play a fundamental role as Bruhat decompositions in the theory of Lie
type groups (see for example, [5]), and more generally in the study of groups with a (B, N )-pair
Lemma 2 and Theorem 4 provide a reduction pathway to the study of nondegenerate triple fac-
(see [4]). With the extra condition that AB ∩ BA = A ∪ B, Higman and McLaughlin [12]
5, A = (1, 2, 3, 4, 5) , B = (1, 3)(2, 5), (1, 2)(3, 5) . Then T = (G, A, B)
torisations T = (G, A, B) with A maximal in G and both A and B core-free in G, as follows.

is a nondegenerate triple factorisation. Moreover, A and B are each contained in a unique max-
showed in their famous paper that the associated coset geometry is a flag-transitive 2-design, and
We call such T primitive. Indeed, if T = (G, A, B) is a nondegenerate triple factorisation with
that G acts primitively on the points of this design (that is, A is a maximal subgroup of G). More
= A4, respectively, and G = CD. Thus the only
coreG(A) = 1, then there exist subgroups H, K such that A ≤ K < H ≤ G, K is maximal
lift T = (G, C, D) of T with C, D maximal is degenerate.

general factorisations of the form G = ABC arise, for example, as Iwasawa decompositions of
in H, G = HB, and the quotient of (H, K, BN ∩ H) modulo coreH(K) is a primitive triple
semisimple Lie groups (see [7, 13]). Here we focus on the case A = C.

factorisation, where N = coreG(K). In particular, if A is maximal in G (so that K = A and
One of the major questions we address in this research is: under what conditions does such a lift
H = G), then T is primitive (see Corollary 1.3 in [1] ).

(G, C, D) of a nondegenerate triple factorisation T = (G, A, B) remain nondegenerate?
From this point of view the basic nondegenerate triple factorisations T = (G, A, B) to study
Special cases of a triple factorisation T = (G, A, B) occur if one of A, B is equal to G, in which
are the primitive ones, that is, those with A and B core-free in G, and A maximal in G. For
case T is said to be trivial, or more generally, if G factorises as G = AB, and here we say that T
For a normal subgroup N of G, the quotient of T modulo N is the triple T /N = (G/N, AN/N,
this study, we may therefore regard G as a primitive permutation group on ΩA with point
is degenerate. If G = AB we call T nondegenerate.

BN/N ). This is always a triple factorisation, and as in the case of lifts, it may be trivial (if N is
stabiliser A, and study intransitive subgroups B with the property of Geometric criterion.

transitive on ΩA or on ΩB), degenerate or nondegenerate. We discuss various necessary and/or
sufficient conditions under which T /N is nondegenerate. In particular, we prove the following
useful fact which translates the problems about triple factorisations to the language of transitive
For each proper subgroup H of a group G, the group G induces a transitive action by right
permutation groups (see Lemma 5.2(c) in [1]).

multiplication on the set ΩH = [G : H] of right cosets of H:
[1] S. H. Alavi, and C. E. Praeger, On triple factorisations of finite groups, (submitted). arXiv:0909.4393v1
Lemma 2. If N ≤ coreG(A)coreG(B), then T is nondegenerate if and only if T /N is nondegen-
[2] B. Amberg, S. Franciosi, and F. de Giovanni, Products of groups, (Oxford University Press, 1992).

(Hg)x := Hgx for all Hg ∈ ΩH, x ∈ G.

[3] M. Bhattacharjee, D. Macpherson, R.G. M¨oller, and P.M. Neumann, Notes on infinite permutation groups, (Hin-
The kernel of this action is the core of H in G, namely coreG(H) = ∩g∈GHg. Thus for a triple
[4] N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6, (Springer-Verlag, 2002).

T = (G, A, B) with A and B proper subgroups, there are two such transitive actions, on ΩA and
[5] R. W. Carter, Simple groups of Lie type, (John Wiley & Sons Inc., 1989).

For a triple factorisation T = (G, A, B), and H ≤ G, the triple T |
[6] J. D. Dixon and B. Mortimer, Permutation groups, (Springer-Verlag, 1996).

H is a triple factorisation, we say that T restricts to H .

[7] A. S. Fedenko and A. I. Shtern, Iwasawa decomposition, In Encyclopaedia of Mathematics, Edited by
Often the study of primitive permutation groups G focuses on the action of its socle, and indeed,
Hazewinkel, Michiel., (Kluwer Academic Publishers, 2001). http://eom.springer.de/I/i053060.htm.

Here we give two necessary and sufficient conditions for T = (G, A, B) to be a triple factorisa-
important (Lie type) triple factorisations arising from the ‘Bruhat decomposition’ often have G
[8] M. Giudici and J. P. James, Factorisations of groups into three conjugate subgroups, (in preparation).

simple. However reduction of triple factorisations to proper normal subgroups is not straightfor-
[9] D. Gorenstein, A class of Frobenius groups, Canad. J. Math. 11 (1959), 39–47.

, On finite groups of the form ABA, Canad. J. Math. 14 (1962), 195–236.

[11] D. Gorenstein and I. N. Herstein, A class of solvable groups, Canad. J. Math. 11 (1959), 311–320.

Let A and B be proper subgroups of a group G, and consider the right coset action of G on
[12] D. G. Higman and J. E. McLaughlin, Geometric ABA-groups, Illinois J. Math. 5 (1961), 382–397.

ΩA := {Ag| g ∈ G}. Set α := A ∈ ΩA. Then T = (G, A, B) is a triple factorization if
Notation 1. Let T = (G, A, B) be a triple factorisation with coreG(A) = 1, and let G act
[13] K. Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558.

and only if the B-orbit αB intersects nontrivially each Gα-orbit in Ω
A and α := A ∈ Ω. Suppose that A < H < G, α ∈ ∆ = αH ,
[14] W. M. Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J.

and that Σ := {∆1 := ∆, . . . , ∆ } with |G : H| = . Using lift, quotient and restriction triple
factorisations, we define the following triples:
To explain the geometric interpretation, for a group G and proper subgroups A, B, call the el-
[15] M. W. Liebeck, C. E. Praeger, and J. Saxl, The maximal factorizations of the finite simple groups and their
• T0 = (G0, A0, B0) = (H∆, A∆, (B ∩ H)∆).

automorphism groups, Mem. Amer. Math. Soc. 86 (1990), no. 432, iv+151.

ements of ΩA and ΩB points and lines, respectively and define a point Ax and a line By to be
incident if and only if Ax ∩ By = ∅. An incident point-line pair (Ax, By) is called a flag. It
1 = (G1, A1, B1) = (GΣ, H Σ, BΣ).

[16] P. M. Neumann and C. E. Praeger, An inequality for tactical configurations, Bull. London Math. Soc. 28 (1996),
follows from this definition that G preserves incidence and acts transitively on the flags of this
[17] C. E. Praeger, Movement and separation of subsets of points under group actions, J. London Math. Soc. (2) 56
geometry. Moreover, T = (G, A, B) is a triple factorisation if and only if any two points lie on
T1 are triple factorisation, so is T0 T1. Such a factorisation is called wreath product of T0 and T1
at least one line (see Lemma 3 in [12]).

which is always nondegenerate (see Theorem 4).

Summer School: Finite Simple Groups and Algebraic Groups: Representations, Geometries and Applications; Berlin: August 31, 2009- September 10, 2009

Source: http://www.math.uni-bielefeld.de/~baumeist/sommerschule/alavi.pdf

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