 Focal subgroup theorem

In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in (Higman 1958) and is the "first major application of the transfer" according to (Gorenstein, Lyons & Solomon 1996, p. 90). The focal subgroup theorem relates the ideas of transfer and fusion such as described in (Grün 1935). Various applications of these ideas include local criteria for pnilpotence and various nonsimplicity criterion focussing on showing that a finite group has a normal subgroup of index p.
Contents
Background
The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of p, the transfer homomorphism, and fusion of elements.
Subgroups
The following three normal subgroups of index a power of p are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) pgroup. Formally, they are kernels of the reflection onto the reflective subcategory of pgroups (respectively, elementary abelian pgroups, abelian pgroups).
 E^{p}(G) is the intersection of all index p normal subgroups; G/E^{p}(G) is an elementary abelian group, and is the largest elementary abelian pgroup onto which G surjects.
 A^{p}(G) (notation from (Isaacs 2008, 5D, p. 164)) is the intersection of all normal subgroups K such that G/K is an abelian pgroup (i.e., K is an index p^{k} normal subgroup that contains the derived group [G,G]): G/A^{p}(G) is the largest abelian pgroup (not necessarily elementary) onto which G surjects.
 O^{p}(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly nonabelian) pgroup (i.e., K is an index p^{k} normal subgroup): G/O^{p}(G) is the largest pgroup (not necessarily abelian) onto which G surjects. O^{p}(G) is also known as the presidual subgroup.
Firstly, as these are weaker conditions on the groups K, one obtains the containments These are further related as:
 A^{p}(G) = O^{p}(G)[G,G].
O^{p}(G) has the following alternative characterization as the subgroup generated by all Sylow qsubgroups of G as q≠p ranges over the prime divisors of the order of G distinct from p.
O^{p}(G) is used to define the lower pseries of G, similarly to the upper pseries described in pcore.
Transfer homomorphism
Main article: Transfer (group theory)The transfer homomorphism is a homomorphism that can be defined from any group G to the abelian group H/[H,H] defined by a subgroup H ≤ G of finite index, that is [G:H] < ∞. The transfer map from a finite group G into its Sylow psubgroup has a kernel that is easy to describe:
 The kernel of the transfer homomorphism from a finite group G into its Sylow psubgroup P has A^{p}(G) as its kernel, (Isaacs 2008, Theorem 5.20, p. 165).
In other words, the "obvious" homomorphism onto an abelian pgroup is in fact the most general such homomorphism.
Fusion
The fusion pattern of a subgroup H in G is the equivalence relation on the elements of H where two elements h, k of H are fused if they are Gconjugate, that is, if there is some g in G such that h = k^{g}. The normal structure of G has an effect on the fusion pattern of its Sylow psubgroups, and conversely the fusion pattern of its Sylow psubgroups has an effect on the normal structure of G, (Gorenstein, Lyons & Solomon 1996, p. 89).
Focal subgroup
If one defines, as in (Gorenstein 1980, p. 246), the focal subgroup of P in G as the intersection P∩[G,G] of the Sylow psubgroup P of the finite group G with the derived subgroup [G,G] of G, then the focal subgroup is clearly important as it is a Sylow psubgroup of the derived subgroup. However, more importantly, one gets the following result:
 There exists a normal subgroup K of G with G/K an abelian pgroup isomorphic to P/P∩[G,G] (here K denotes A^{p}(G)), and
 if K is a normal subgroup of G with G/K an abelian pgroup, then P∩[G,G] ≤ K, and G/K is a homomorphic image of P/P∩[G,G], (Gorenstein 1980, Theorem 7.3.1, p. 90).
One can define, as in (Isaacs 2008, p. 165) the focal subgroup of H with respect to G as:
 Foc_{G}(H) = ⟨ x^{−1} y  x,y in H and x is Gconjugate to y ⟩.
This focal subgroup measures the extent to which elements of H fuse in G, while the previous definition measured certain abelian pgroup homomorphic images of the group G. The content of the focal subgroup theorem is that these two definitions of focal subgroup are compatible.
Statement of the theorem
The focal subgroup of a finite group X with Sylow psubgroup P is given by:
 P∩[G,G] = P∩A^{p}(G) = P∩ker(v) = Foc_{G}(P) = ⟨ x^{−1} y  x,y in P and x is Gconjugate to y ⟩
where v is the transfer homomorphism from G to P/[P,P], (Isaacs 2008, Theorem 5.21, p. 165).
History and generalizations
This connection between transfer and fusion is credited to (Higman 1958),^{[1]} where, in different language, the focal subgroup theorem was proved along with various generalizations. The requirement that G/K be abelian was dropped, so that Higman also studied O^{p}(G) and the nilpotent residual γ_{∞}(G), as so called hyperfocal subgroups. Higman also did not restrict to a single prime p, but rather allowed πgroups for sets of primes π and used Philip Hall's theorem of Hall subgroups in order to prove similar results about the transfer into Hall πsubgroups; taking π = {p} a Hall πsubgroup is a Sylow psubgroup, and the results of Higman are as presented above.
Interest in the hyperfocal subgroups was renewed by work of (Puig 2000) in understanding the modular representation theory of certain well behaved blocks. The hyperfocal subgroup of P in G can defined as P∩γ_{∞}(G) that is, as a Sylow psubgroup of the nilpotent residual of G. If P is a Sylow psubgroup of the finite group G, then one gets the standard focal subgroup theorem:
 P∩γ_{∞}(G) = P∩O^{p}(G) = ⟨ x^{−1} y : x,y in P and y = x^{g} for some g in G of order coprime to p ⟩
and the local characterization:
 P∩O^{p}(G) = ⟨ x^{−1} y : x,y in Q ≤ P and y = x^{g} for some g in N_{G(Q)} of order coprime to p ⟩.
This compares to the local characterization of the focal subgroup as:
 P∩A^{p}(G) = ⟨ x^{−1} y : x,y in Q ≤ P and y = x^{g} for some g in N_{G}(Q) ⟩.
Puig is interested in the generalization of this situation to fusion systems, a categorical model of the fusion pattern of a Sylow psubgroup with respect to a finite group that also models the fusion pattern of a defect group of a pblock in modular representation theory. In fact fusion systems have found a number of surprising applications and inspirations in the area of algebraic topology known as equivariant homotopy theory. Some of the major algebraic theorems in this area only have topological proofs at the moment.
Other characterizations
Various mathematicians have presented methods to calculate the focal subgroup from smaller groups. For instance, the influential work (Alperin 1967) develops the idea of a local control of fusion, and as an example application shows that:
 P ∩ A^{p}(G) is generated by the commutator subgroups [Q, N_{G}(Q)] where Q varies over a family C of subgroups of P
The choice of the family C can be made in many ways (C is what is called a "weak conjugation family" in (Alperin 1967)), and several examples are given: one can take C to be all nonidentity subgroups of P, or the smaller choice of just the intersections Q = P ∩ P^{g} for g in G in which N_{P}(Q) and N_{Pg}(Q) are both Sylow psubgroups of N_{G}(Q). The latter choice is made in (Gorenstein 1980, Theorem 7.4.1, p. 251). The work of (Grün 1935) studied aspects of the transfer and fusion as well, resulting in Grün's first theorem:
 P ∩ A^{p}(G) is generated by P∩[N,N] and P ∩ [Q, Q] where N = N_{G}(P) and Q ranges over the set of Sylow psubgroups Q = P^{g} of G (Gorenstein 1980, Theorem 7.4.2, p. 252).
Applications
The textbook presentations in (Rose 1978, pp. 254–264), (Isaacs 2008, Chapter 5), (Hall 1959, Chapter 14), (Suzuki 1986, §5.2, pp. 138–165), all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of splitting called pnilpotence.
During the course of the Alperin–Brauer–Gorenstein theorem classifying finite simple groups with quasidihedral Sylow 2subgroups, it becomes necessary to distinguish four types of groups with quasidihedral Sylow 2subgroups: the 2nilpotent groups, the Qtype groups whose focal subgroup is a generalized quaternion group of index 2, the Dtype groups whose focal subgroup a dihedral group of index 2, and the QDtype groups whose focal subgroup is the entire quasidihedral group. In terms of fusion, the 2nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the Qtype have 2 classes of involutions and one class of cyclic subgroup of order 4; the QDtype have one class each of involutions and cyclic subgroups of order 4. In other words, finite groups with quasidihedral Sylow 2subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns. The explicit lists of groups with each fusion pattern are contained in (Alperin, Brauer & Gorenstein 1970).
Notes
 ^ The focal subgroup theorem and/or the focal subgroup is due to (Higman 1958) according to (Gorenstein, Lyons & Solomon 1996, p. 90), (Rose 1978, p. 255), (Suzuki 1986, p. 141); however, the focal subgroup theorem as stated there and here is quite a bit older and already appears in textbook form in (Hall 1959, p. 215). There and in (Puig 2000) the ideas are credited to (Grün 1935); compare to (Grün 1935, Satz 5) in the special case of pnormal groups, and the general result in Satz 9 which is in some sense a refinement of the focal subgroup theorem.
References
 Alperin, J. L. (1967), "Sylow intersections and fusion", Journal of Algebra 6: 222–241, doi:10.1016/00218693(67)900051, ISSN 00218693, MR0215913
 Alperin, J. L.; Brauer, R.; Gorenstein, D. (1970), "Finite groups with quasidihedral and wreathed Sylow 2subgroups.", Transactions of the American Mathematical Society (American Mathematical Society) 151: 1–261, doi:10.2307/1995627, ISSN 00029947, MR0284499
 Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 9780828403016, MR81b:20002
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1996), The classification of the finite simple groups. Number 2. Part I. Chapter G, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821803905, MR1358135, https://www.ams.org/online_bks/surv402
 Grün, Otto (1935), "Beiträge zur Gruppentheorie. I." (in German), Journal für Reine und Angewandte Mathematik 174: 1–14, ISSN 00754102, Zbl 0012.34102, http://resolver.sub.unigoettingen.de/purl?GDZPPN002173409
 Hall, Marshall, Jr. (1959), The theory of groups, New York: Macmillan, MR0103215
 Higman, Donald G. (1953), "Focal series in finite groups", Canadian Journal of Mathematics 5: 477–497, ISSN 0008414X, MR0058597, http://www.cms.math.ca/cjm/v5/p477
 Puig, Lluis (2000), "The hyperfocal subalgebra of a block", Inventiones Mathematicae 141 (2): 365–397, doi:10.1007/s002220000072, ISSN 00209910, MR1775217
 Rose, John S. (1994) [1978], A Course in Group Theory, New York: Dover Publications, ISBN 9780486681948, MR0498810
 Suzuki, Michio (1986), Group theory. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 248, Berlin, New York: SpringerVerlag, ISBN 9780387109169, MR815926
Categories: Pgroups
 Theorems in algebra
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