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Adolph Hurwitz 1859-1919 Adolph Hurwitz Timeline • 1859 born • 1881 doctorate under Felix Klein • Frobenius’ successor, ETH Zurich, 1892 • Died 1919, leaving many unpublished notebooks. • George Polya drew attention to the contents. • 1926 Fritz Gassmann (double s, double n, no r) published one set of Hurwitz’s notes followed by Gassmann’s interpretation of what Hurwitz meant. In Gassmann’s paper, the following group-theoretic condition appeared for two subgroups H1, H2 of a group G: |gG ∩ H1| = |gG ∩ H2| for every conjugacy class gG in G. A group G and subgroups H1, H2 form a Gassmann triple when Gassmann’s Criterion holds: (I) |gG ∩ H1| = |gG ∩ H2| for every conjugacy class gG in G. Let χi(g) = number of cosets of G/Hi (i = 1,2) fixed by left-multiplication by g. Reformulation: Gassmann’s criterion (1) holds if and only if (2) for all g in G. χ1(g) = χ2(g) Reformulation: (1) holds iff (3) Q[G/H1] and Q[G/H2] are isomorphic Q[G]-modules. Reformulation: (1) holds iff (4) There is bijection H1 H2 which is a local conjugation in G. • When any of these criteria hold, then (G:H1) = (G:H2). • Conjugate subgroups H1, H2 of G are always Gassmann equivalent; this is the case of trivial Gassmann equivalence. We are interested in nontrivial Gassmann equivalent subgroups. Applications: Gassmann triples (G, H1, H2) can be used to produce • pairs of arithmetically equivalent number fields (identical zeta functions); • pairs of isospectral riemannian manifolds; • pairs of nonisomorphic finite graphs with identical Ihara zeta functions; I thought it would be interesting to collect some results about Gassmann triples. Exercise: Translate each of the statement below into a statement about arithmetically equivalent number fields, about isospectral manifolds, and about graphs with the same Ihara zeta functions. Organization: 1. Small index 2. Solvable groups 3. Prime index 4. Index p2, p prime 5. Index 2p+2, p an odd prime. 6. Beaulieu’s construction 7. Involutions with many fixed points 1. Small index: (P, 1978, de Smit-Bosma, 2005) Number of faithful, nontrivial Gassmann triples of index (G:H1) = n. Index n Number of triples of Index n ≤ 6 0 7 1 8 2 9, 10 0 11 1 12 6 13 1 14 4 15 4 2. Solvable Groups The Lenstra-de Smit Theorem (1998): Let n be a positive integer. Then the following are equivalent: 1. There exists a nontrivial solvable Gassmann triple of index n 2. There are prime numbers p, q, r (possibly equal) with pqr | n and p | q(q-1) 3. Prime Index Feit’s Theorem (1980): Let (G, H1, H2) be a nontrivial Gassmann triple of prime index n=p. Then either p = 11 or p = (qk – 1) / (q – 1) for some prime power q and some k ≥ 3. 4. Index p2, p a prime Guralnick’s Theorem ( 1983): Let p be a prime. There is a nontrivial Gassmann triple of index p2 iff pe = (qk –1) / (q-1) for some e≤2, k≥3, and some prime-power q. 5. Index 2p+2, p an odd prime. de Smit’s Construction, (2003): For every odd prime p there is a nontrivial Gassmann triple of index n=2p+2. 6. Beaulieu’s Construction Beaulieu’s Theorem (1996): Let (G, H, H') be a faithful, nontrivial triple of index n having no automorphism σ in Aut(G) taking H to H'. Let π (resp. π') : G Sn be the permutation representations coming from left translation of G on G/H (resp. of G on G/H'). Set G1= Sn , H1 = π(G), and H1′ = π′(G). Then (G1, H1, H1′) is a faithful nontrivial triple of index > n with no outer automorphism taking H1 to H1′ . ************************************************************************************ Iteration gives infinitely many triples arising canonically from the first triple. 7. Involutions with many fixed-points The Chinburg-Hamilton-Long-Reid Theorem (2008): Every Gassmann triple (G, H1, H2) of index n, and containing an involution δ with χ1(δ) = n-2, is trivial. One Interpretation: If K it a number field of degree n over Q having exactly n-2 real embeddings, then K is determined (up to isomorphism) by its zeta function.