1. Adolph Hurwitz 1859-1919

2. 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Timeline 1859 born 1881 doctorate under Felix Klein Frobenius’ successor, ETH Zurich, 1892 Died 1919, leaving many unpublished notebooks.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.

3. In Gassmann’s paper, the following group-theoretic condition appeared for two subgroups H1, H2 of a group G: |g G ∩ H1| = |g G ∩ H2| for every conjugacy class g G in G.

4. A group G and subgroups H1, H2 form a Gassmann triple when Gassmann’s Criterion holds: (I) |g G ∩ H1| = |g G ∩ H2| for every conjugacy class g G in G. Let χ i (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) χ 1 (g) = χ 2 (g) for all g in G.

5. 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.

6. 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.

7. Organization: 1.Small index 2.Solvable groups 3.Prime index 4.Index p 2, p prime 5.Index 2p+2, p an odd prime. 6.Beaulieu’s construction 7.Involutions with many fixed points

8. 1.Small index: (P, 1978, de Smit-Bosma, 2005) Number of faithful, nontrivial Gassmann triples of index (G:H1) = n. Index nNumber of triples of Index n ≤ 6 0 7 1 8 2 9, 10 0 11 1 12 6 13 1 14 4 15 4

9. 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)

10. 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 = (q k – 1) / (q – 1) for some prime power q and some k ≥ 3.

11. 4.Index p 2, p a prime Guralnick’s Theorem ( 1983): Let p be a prime. There is a nontrivial Gassmann triple of index p 2 iff p e = (q k –1) / (q-1) for some e≤2, k≥3, and some prime-power q.

12. 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.

13. 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  S n be the permutation representations coming from left translation of G on G/H (resp. of G on G/H'). Set G1= S n, 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.

14. 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 χ 1 (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.