# Existence of certain Hall-subgroups based on knowledge of the degrees of irreducible characters

I am trying to find out what results are already out there in this direction. For example, from the Ito-Michler theorem, if $\rho(G)$ is the set of prime divisors of irreducible characters of $G$ and either $\rho(G)\subseteq \pi$ or $|\rho(G)\cap\pi|\leq 1$ then we are guaranteed the existence of a $\pi$-Hall subgroup. Also, there is a result by J. Thompson that if some prime $p$ divides the degrees of all non-linear irreducible characters of a finite groups, then that group has a normal $p$-complement.

Are there other results like these, which give a relations between some set connected to the degrees of irreducible characters of a group and a set $\pi$ of prime numbers, such that we are guaranteed the existence of a $\pi$-Hall subgroup?

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Background:

(Ito–Michler) p does not divide the order of the degree of any (absolutely) irreducible (ordinary) character of G iff G has a normal, abelian, Sylow p-subgroup.

For any subset π of ρ(G)′, it follows that G has a normal, abelian, Hall π-subgroup, and hence also a Hall π′-subgroup. Flipping the roles of π and π′ gives your first existence claim.

Chapter 14 of Isaacs's Character Theory textbook contains this and further results, which I assume are already known. For instance, the classical:

(Blichfeldt) if G has a faithful (possibly reducible, ordinary) character of degree n, then for any π consisting of primes p > n, G has an abelian, Hall π-subgroup.

Blichfeldt's paper below contains the result that G has an abelian, normal, Hall π-subgroup for π a collection of primes p > (n-1)(2n+1). This was refined to:

(Feit–Thompson) if G has a faithful (possibly reducible, ordinary) character of degree n, then for any π consisting of primes p > 2n+1, G has an abelian, normal, Hall π-subgroup.

The more detailed information alluded to in Isaacs's textbook is likely Winter's 1964 paper below. It gives a complete list of groups with non-normal Sylow p-subgroups for p > (2n+1)/3. A non-abelian but normal Sylow π-subgroup is implied by the following:

(Feit) if G has a a faithful, irreducible (ordinary) character of degree p-2 ≥3, then either the Sylow p-subgroup of G is normal, G is a direct product of SL(2,p) and an abelian group, or G is a direct product of 3.Alt(6) and abelian group.

Now each of these results required faithful characters. If you are OK with that, then Berkovich (MR1189113) has a Thompson like hypothesis and an Ito like conclusion (normal Hall π-subgroup containing the nilpotent residual), but it looks at π(G/Z(χ)) too.

The Ito–Michler result is also valid in some sense over finite fields:

(Manz) Every irreducible, p-modular character degree is coprime to p iff G has a normal Sylow p-subgroup.

(Manz–Wolf) If G is p-solvable and every irreducible, p-modular character degree is coprime to q, then Oq′(G), the subgroup generated by the Sylow q-subgroups, is solvable of q-length at most 2 (π-length roughly measures how far away from having a normal Hall π-subgroup you are).

(Navarro, et al.) has generalized both the Ito and Thompson results to blocks in MR1810119, MR1956546, and MR2159761.

A more recent article by (Matterei) gives a reasonable survey of results. The way he rephrases results may be quite interesting to you. He also gives the Carter–Hawkes (MR1199667) results that continue the ideas of Thompson. A recent paper of Dolfi et. al (MR2469367) derives Ito-like results using the orders of a non-vanishing elements g in G, such that χ(g) ≠ 0 for any any ordinary, irreducible character χ and derives Ito's and Thompson's results as corollaries.

Blichfeldt, H. F. "On the order of linear homogeneous groups." Trans. AMS. 4 (1903), 387–397. JFM 34.0176.02 JSTOR 1986408

Feit, Walter; Thompson, John G. "Groups which have a faithful representation of degree less than (p-1)/2". Pacific J. Math. 11 (1961), 1257–1262. MR133373 euclid.pjm/1103036911

Winter, David L. "Finite groups having a faithful representation of degree less than (2p+1)/3." Amer. J. Math. 86 (1964), 608–618. MR183788 DOI: 10.2307/2373026

Feit, Walter. "On finite linear groups. II." J. Algebra 30 (1974), 496–506. MR357634 DOI: 10.1016/0021-8693(74)90220-8

Manz, Olaf. "On the modular version of Ito's theorem on character degrees for groups of odd order." Nagoya Math. J. 105 (1987), 121–128. MR881011 euclid.nmj/1118780642

Manz, Olaf; Wolf, Thomas R. "Brauer characters of q′-degree in p-solvable groups." J. Algebra 115 (1988), no. 1, 75–91. MR937602 DOI: 10.1016/0021-8693(88)90283-9

Mattarei, Sandro. "Retrieving information about a group from its character degrees or from its class sizes." Proc. Amer. Math. Soc. 134 (2006), no. 8, 2189–2195. MR2213690 DOI: 10.1090/S0002-9939-06-08274-8

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Thanks, this should give me plenty of good articles to read. – Tobias Kildetoft Apr 18 '10 at 9:57