On a theorem of Jordan

By Jean-Pierre Serre. 2003

157 citations recorded by the source corpus.

The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology. 1. Statements 1.1. Number theory. Let f = ∑n m=0 amx m be a polynomial of degree n, with coefficients in Z. If p is prime, let Np(f) be the number of zeros of f in Fp = Z/pZ. Theorem 1. Assume (i) n ≥ 2, (ii) f is irreducible in Q[x]. Then (a) There are infinitely many p’s with Np(f) = 0. (b) The set P0(f) of p’s with Np(f) = 0 has a density c0 = c0(f) which is > 0. [Recall that a subset P of the set of primes has density c if lim X→∞ number of p ∈ P with p ≤ X π(X) = c, where π(X) is as usual the number of primes ≤ X .] Moreover, Theorem 2. With the notation of Theorem 1, one has c0(f) ≥ 1 n , with strict inequality if n is not a power of a prime. Example. Let f = x + 1. One has p ∈ P0(f) if and only if p ≡ −1 (mod 4); this set is well-known to have density 1/2. We shall see more interesting examples in §5. 1.2. Topology. Let S1 be a circle. Let f : T → S be a finite covering of a topological space S. Assume: (i) f has degree n (i.e. every fiber of f has n elements), with n ≥ 2, (ii) T is arcwise connected and not empty. Theorem 3. There exists a continuous map φ : S1 → S which cannot be lifted to the covering T (i.e. there does not exist any continuous map ψ : S1 → T such that φ = f ◦ ψ). Received by the editors March 1, 2003. 2000 Mathematics Subject Classification. Primary 06-XX, 11-XX, 11F11. This text first appeared in Math Medley 29 (2002), 3–18. The writing was done with the help of Heng Huat Chan. c ©2002 Singapore Mathematical Society. Reprinted with permission.

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

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Citations in the corpus

Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements (2009, background): “ortion of derangements in G. By an old theorem of Jordan, it follows that δ(G,Ω) >0 if |Ω| >1. By a much more recent (but still elementary) theorem [CC], δ(G,Ω) ≥ 1/|Ω| for |Ω| >1. See Serre [Se] for many applications of Jordan’s theorem. See [GW] for applications of better bounds of δ(G,Ω)”
Commutative quotients of finite W-algebras (2008, unknown): “As explained in [36], for instance, it follows from Burnside’s Lemma and Chebotarev’s Density Theorem that”
On fixed points of permutations (2007, methodology): “For further discussion, including a topological application of Jordan’s theorem, see [63].”
Between primitive and 2-transitive: Synchronization and its friends (2015, background): “The non-existence in the second case also follows from an old result of Jordan (see Serre [95]), according to which a finite transitive permutation group of degree greater than 1 contains a fixed-point-free element.”
Hilbert's irreducibility theorem and the larger sieve (2010, unknown): “≤B,Gt6= G}|≤ X M n t∈On k−Ωf : ||t||≤B,Gt⊆ [ g∈G gMg−1 o where the sum is over representatives of the conjugacy classes of maximal subgroups of G.Deﬁne δ(G,M) := | S g∈GgMg −1|/|G|. By Jordan’s lemma [Ser03], we know that S g∈GgMg −1 is a proper subset of G(equivalently δ(G,M) <1). Applying the b”