Conjugacy of nilpotent injectors in soluble groups

Hello,

I was wondering if anyone is aware of an elementary proof of the claim in the title, assuming the existence of nilpotent injectors in soluble groups. By elementary I mean a proof that does not involve any recourse to Fitting classes etc.

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You probably should state what definition of nilpotent injector you want to use, since even specialists use various different definitions. – Geoff Robinson May 19 '11 at 6:31
You may want to look at Wehrfritz's textbook Finite Groups, chapter 7. He discussed Fischer groups from a few standpoints, and at least handles Carter subgroups (the dual) in a very elementary manner. Also compared to say Doerk–Hawkes, his Fitting theory is quite a bit more elementary, as in, easy to learn the first time. Fischer's original 1966 characterization of nilpotent-injectors predates fitting classes by a tiny bit. As Geoff Robinson mentions though, it uses a different definition. The equivalence of the two (or three) definitions is on page 100 of Wehrfritz's textbook. – Jack Schmidt May 19 '11 at 16:19
@Jack: Thank you, I will have a look. – user13040 May 19 '11 at 22:06

I take a nilpotent injector of a finite solvable group $G$ to be a nilpotent subgroup $M$ of $G$ such that $M \cap N$ is a maximal nilpotent normal subgroup of $N$ whenever $N$ is subnormal in $G$. Assuming existence of $M$ , I think uniqueness up to conjugacy follows inductively. Notice that $M \cap H$ is a nilpotent injector of $H$ whenever $H$ is normal in $G.$

We may suppose that $Z(G) = 1$. Now let $p$ be a prime divisor of $|F(G)|$. Since $F(G) \leq M$, we have $O_{p'}(M) \leq C_{G}(O_{p}(G)).$ Thus $O_{p'}(M) = O_{p'}(L)$, where $L = M \cap C_{G}(O_{p}(G))$ is a nilpotent injector of $C_{G}(O_{p}(G))$. For notice that $O_{p'}(L) \lhd M$ so that $O_{p'}(L) \leq O_{p'}(M)$, while $O_{p'}(M) \leq M \cap C_{G}(O_{p}(G)) =L$ and $O_{p'}(M) \leq O_{p'}(L)$.

Now $L$ is unique up to conjugacy within $C_{G}(O_{p}(G))$, so certainly unique up to conjugacy in $G$. Hence $O_{p'}(M)= O_{p'}(L)$ is unique up to conjugacy within $G$. By maximality as a nilpotent subgroup, $M = P \times O_{p'}(M)$, where $P$ is a Sylow $p$-subgroup of $C_{G}(O_{p'}(M)).$ Hence we see that $M$ is unique up to conjugacy. in $G$.

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Thank you for your answer. I take a nilpotent injector of a soluble group to be a nilpotent subgroup containing the Fitting subgroup and maximal (under inclusion) with that property. – user13040 May 19 '11 at 22:12
In that case, you are really looking for an elementary proof of Fischer's theorem, as mentioned by Jack Schmidt. Well, as Jack mentions, Fischer gave this proof before Fitting classes were invented, so Fischer's own proof is elemntary in the sense you wanted. Once we know that these "Fischer subgroups" are unique up to conjugacy, they must be nilpotent injectors in the above sense. But then I am slightly perplexed by the question, since existence is not an issue. – Geoff Robinson May 20 '11 at 0:15
Indeed it is not; I should have expressed myself in a better way. Regarding Fischer's proof, are you referring to "Klassen konjugierter Untergruppen in endlichen auflösbaren Gruppen"? If so, then the reason I didn't read it in the first place is because I don't know any german and was unable to find a 'reproduction' in english. To be honest I am out of my league here; I encountered this as a problem in Isaacs' 'Finite Group Theory' book (specifically 3C.8 in page 91) and, while mathoverflow is not meant to be a problem-solver, I thought it wouldn't be completely inappropriate to ask. – user13040 May 20 '11 at 2:28
Well, Jack Schmidt has now provided you with references in English. I have never read Fischer's original paper, but have cooked a proof by myself in the past, which I assume was the same as Fischer's. – Geoff Robinson May 20 '11 at 6:55
I contacted Isaacs himself about this and he said that his 'favorite proof' is something more general: if the intersection of two maximal nilpotent subgroups of a finite group G contains its own centralizer, then these subgroups are conjugate. I think the result Isaacs mentions is due to Hans Lausch in his paper "Conjugacy classes of maximal nilpotent subgroups" from 1984 springerlink.com/content/p106nl14504n8648. I would be in debt to anyone who might consider sending this paper to the following address: ucabsai@live.ucl.ac.uk, as I'm having trouble accessing springer remotely. – user13040 May 22 '11 at 2:51

Another good reference here is "Injectors and Normal Subgroups of Finite Groups" by Avinoam Mann. Israel Journal of Mathematics, Vol 9.

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