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I'm interested in results about functorially-defined subgroups (in a loose sense), especially in the non-abelian case, and would like to know about references I may have missed.

The question, it seems, comes up in its simplest form when noticing a number of common subgroups (the center, commutator subgroup, Frattini subgroup, etc) are characteristic. The characteristicity can be justified by the fact that the object mappings that define of those subgroups give rise to subfunctors of the identity functor, on the core category of Grp.

Hence I'm interested in functors F in Grp (or a carefully chosen sub-category) such that

∀ A F(A) ⊆ A and ∀ A,B ∀ f ∈ hom(A, B), f(F(A)) ⊆ F(B)

Does that ring a bell ?

The topic was mentioned a couple of months ago on Mathoverflow, and can be traced back to (at least) 1945, where Saunders MacLane explains it in some detail in the third chapter of A General Theory of Natural Equivalences.

In between, it seems that those functors have been baptised radicals, pre-radicals or subgroup functorials, and studied mostly in the framework of ring theory, notably by A. Kurosh. Among a number of not-so-recent (and therefore quite-hard-to-find) papers mostly dealing with rings, semigroups, or abelian groups, I came across a single reference mentioning the non-abelian case, by B.I. Plotkin : Radicals in groups, operations on classes of groups, and radical classes. Connections seem have been made with closure operators¹, but do not focus much on Grp.

  • Do you have ideas of connections from those functors to other parts of algebra or category theory, other than (pre-)radicals ?
  • Do you have some pointers to material I may have missed, specially if they mention non-abelian groups ?

¹: Categorical structure of closure operators with applications to topology By N. Dikranjan, Walter Tholen, p.51

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Do these functors need to be only on groups? For example, the concept of an adic $A$-algebra (for $A$ a ring) is a ring $B$ with a map $B\rightarrow A$ such that the kernel is a nilpotent ideal. The category of these has a functor (taking the kernel of the map to $A$ for each ring) like you describe. – Inna Feb 6 '10 at 1:32
up vote 3 down vote accepted

After some research, I asked on the Group-pub mailing list, where very knowledgeable people roam (the University of Bath, which hosts the mailing-list, also hosted the 'Groups St Andrews' conference in 2009). Jan Krempa pointed me to a recent seminar on radicals, that included a very useful survey paper: B. J. Gardner, Kurosh-Amitsur radicals of groups: something for overyone?

Its lists of references is a goldmine, particularly the included book by the same author (Radical Theory). It includes general radical theory that applies to the non-commutative case better than his later book (written with R. Wiegandt :Radical theory of rings).

I also got a nice answer from Mike Newman (presumptively this one), who told me:

** An early interest occurs in

MR0002876 (2,125i) Hall, P. Verbal and marginal subgroups. J. Reine Angew. Math. 182, (1940). 156--157.

This is a brief report of a lecture given at a meeting in Goettingen in 1939.

** There is recent work which hinges on these sorts of ideas in

MR2276769 (2008f:20052) Nikolov, Nikolay(4-OXNC); Segal, Dan(4-OXAS) On finitely generated profinite groups. I. Strong completeness and uniform bounds. (English summary) Ann. of Math. (2) 165 (2007), no. 1, 171--238. 20E18 (20E32 20F12)

The link with verbal and marginal subgroups was already hinted at in the previous mathoverflow question I mentioned.

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