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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• Died 1919, leaving many unpublished notebooks.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.