1
The Large Davenport Constant for Non-Abelian
Groups
David Grynkiewicz
research in collaboration with A. Geroldinger
Karl-Franzens-Universität Graz
January 10, 2013
1 Supported
in part by the Austrian Science Foundation under grant P21576-N18.
Sequences
Sequences
Let G be a multiplicatively written finite group
Sequences
Let G be a multiplicatively written finite group
Let F(G ) denote the free abelian monoid with basis G and monoid
operation ·
Sequences
Let G be a multiplicatively written finite group
Let F(G ) denote the free abelian monoid with basis G and monoid
operation ·
An element S ∈ F(G ) is called an (unordered) sequence, and has the
form
S = g1 · . . . · g`
Sequences
Let G be a multiplicatively written finite group
Let F(G ) denote the free abelian monoid with basis G and monoid
operation ·
An element S ∈ F(G ) is called an (unordered) sequence, and has the
form
S = g1 · . . . · g`
Example
S = 2 · 2 · 5 · −1 = 2 · −1 · 5 · 2 ∈ F (Z/7Z)∗
is a sequence of nonzero residue classes modulo 7 consisting of two terms
equal to 2, one term equal to 5, and one term equal to −1.
Product-One Sequences
Product-One Sequences
If there exist an ordering of the terms of a sequence S ∈ F(G ), say as
given by the indexing
S = g1 · . . . · g` ,
such that
`
Y
gi = 1,
i=1
then we say that S is a product-one sequence.
Product-One Sequences
If there exist an ordering of the terms of a sequence S ∈ F(G ), say as
given by the indexing
S = g1 · . . . · g` ,
such that
`
Y
gi = 1,
i=1
then we say that S is a product-one sequence.
Example
S = 2 · 2 · 5 · −1 ∈ F (Z/7Z)∗
Then (22 )(5)(−1) = −20 ≡ 1 mod 7, so S is a product-one sequence.
Block Monoids
Block Monoids
The block monoid of G is defined as
B(G ) = {S ∈ F(G ) : S is a product-one sequence}
Block Monoids
The block monoid of G is defined as
B(G ) = {S ∈ F(G ) : S is a product-one sequence}
Clearly this is a monoid, since if S and T are both product-one, then so
is S · T
Block Monoids
The block monoid of G is defined as
B(G ) = {S ∈ F(G ) : S is a product-one sequence}
Clearly this is a monoid, since if S and T are both product-one, then so
is S · T since if S = g1 · . . . · g` and T = h1 · . . . · h`0 with
`
Y
i=1
0
gi =
`
Y
hi = 1,
i=1
then S · T = g1 · . . . · g` · h1 · . . . · h`0 with
`
`0
Y
Y
hi = 12 = 1.
gi
i=1
i=1
The Davenport Constant
The Davenport Constant
Definition
Let d(G ) denote the maximal length of a sequence S ∈ F(G ) that has
no nontrivial product-one subsequence.
The Davenport Constant
Definition
Let d(G ) denote the maximal length of a sequence S ∈ F(G ) that has
no nontrivial product-one subsequence. Thus d(G ) + 1 is the minimal
length of sequence needed to guarantee a nontrivial product-one
subsequence.
The Davenport Constant
Definition
Let d(G ) denote the maximal length of a sequence S ∈ F(G ) that has
no nontrivial product-one subsequence. Thus d(G ) + 1 is the minimal
length of sequence needed to guarantee a nontrivial product-one
subsequence. This is the small Davenport Constant
The Davenport Constant
Definition
Let d(G ) denote the maximal length of a sequence S ∈ F(G ) that has
no nontrivial product-one subsequence. Thus d(G ) + 1 is the minimal
length of sequence needed to guarantee a nontrivial product-one
subsequence. This is the small Davenport Constant
Definition
Let D(G ) denote the maximal length of an atom in B(G ),
The Davenport Constant
Definition
Let d(G ) denote the maximal length of a sequence S ∈ F(G ) that has
no nontrivial product-one subsequence. Thus d(G ) + 1 is the minimal
length of sequence needed to guarantee a nontrivial product-one
subsequence. This is the small Davenport Constant
Definition
Let D(G ) denote the maximal length of an atom in B(G ), this is a
product-one sequence S ∈ F(G ) that does not have a factorization
S = T1 · T2
with T1 and T2 both nontrivial product-one sequences.
The Davenport Constant
Definition
Let d(G ) denote the maximal length of a sequence S ∈ F(G ) that has
no nontrivial product-one subsequence. Thus d(G ) + 1 is the minimal
length of sequence needed to guarantee a nontrivial product-one
subsequence. This is the small Davenport Constant
Definition
Let D(G ) denote the maximal length of an atom in B(G ), this is a
product-one sequence S ∈ F(G ) that does not have a factorization
S = T1 · T2
with T1 and T2 both nontrivial product-one sequences. This is the large
Davenport Constant
The Davenport Constant in Context
The Davenport Constant in Context
I
The study of D(G ) is one of the older and more difficult questions in
Combinatorial Number Theory.
The Davenport Constant in Context
I
The study of D(G ) is one of the older and more difficult questions in
Combinatorial Number Theory.
I
Exact value of D(G ) known for rank 2 abelian groups and abelian
p-groups.
The Davenport Constant in Context
I
The study of D(G ) is one of the older and more difficult questions in
Combinatorial Number Theory.
I
Exact value of D(G ) known for rank 2 abelian groups and abelian
p-groups.
I
Plays an important role in the study of non-unique factorizations of
elements in a krull monoid.
The Davenport Constant in Context
I
The study of D(G ) is one of the older and more difficult questions in
Combinatorial Number Theory.
I
Exact value of D(G ) known for rank 2 abelian groups and abelian
p-groups.
I
Plays an important role in the study of non-unique factorizations of
elements in a krull monoid.
I
Standard transfer homomorphism machinery often reduces the
general case to the study of block monoids over the class group.
The Davenport Constant in Context
I
The study of D(G ) is one of the older and more difficult questions in
Combinatorial Number Theory.
I
Exact value of D(G ) known for rank 2 abelian groups and abelian
p-groups.
I
Plays an important role in the study of non-unique factorizations of
elements in a krull monoid.
I
Standard transfer homomorphism machinery often reduces the
general case to the study of block monoids over the class group.
For G abelian, D(G ) = β(G ) is equal to the Noether number in
Invariant Theory.
I
Easy Facts
Easy Facts
I
If G is abelian, then d(G ) + 1 = D(G ).
Easy Facts
I
If G is abelian, then d(G ) + 1 = D(G ).
I
In general, d(G ) + 1 ≤ D(G )
Easy Facts
I
If G is abelian, then d(G ) + 1 = D(G ).
I
In general, d(G ) + 1 ≤ D(G )
I
and D(G ) ≤ |G |.
Easy Facts
I
If G is abelian, then d(G ) + 1 = D(G ).
I
In general, d(G ) + 1 ≤ D(G )
I
and D(G ) ≤ |G |.
I
But d(G ) + 1 < D(G ) is possible for G non-abelian.
Easy Facts
I
If G is abelian, then d(G ) + 1 = D(G ).
I
In general, d(G ) + 1 ≤ D(G )
I
and D(G ) ≤ |G |.
I
But d(G ) + 1 < D(G ) is possible for G non-abelian.
I
d(G ) + 1 < β(G ) is also possible for G non-abelian.
Easy Facts
I
If G is abelian, then d(G ) + 1 = D(G ).
I
In general, d(G ) + 1 ≤ D(G )
I
and D(G ) ≤ |G |.
I
But d(G ) + 1 < D(G ) is possible for G non-abelian.
I
d(G ) + 1 < β(G ) is also possible for G non-abelian.
I
But no known counter-examples to d(G ) + 1 ≤ β(G ) ≤ D(G ).
The Small Davenport Constant: Prior Results
The Small Davenport Constant: Prior Results
Theorem (Olson and White)
If G is non-cyclic, then
1
|G |,
2
with equality if G contains a cyclic, index 2 subgroup.
d(G ) ≤
The Small Davenport Constant: Prior Results
Theorem (Olson and White)
If G is non-cyclic, then
1
|G |,
2
with equality if G contains a cyclic, index 2 subgroup.
d(G ) ≤
Theorem (Bass)
If G ∼
= Fpq is isomorphic to the non-abelian group of order pq, where
p < q are primes, then
d(G ) + 1 = q + p − 1.
The Large Davenport Constant: New Results
The Large Davenport Constant: New Results
Recall that d(G ) + 1 ≤ D(G ) with equality for G abelian
The Large Davenport Constant: New Results
Recall that d(G ) + 1 ≤ D(G ) with equality for G abelian
Theorem
Let G be a finite group with G 0 = [G , G ] ≤ G its commutator subgroup.
Then
D(G ) ≤ d(G ) + 2|G 0 | − 1
with equality if and only if G is abelian
The Large Davenport Constant: New Results
The Large Davenport Constant: New Results
Recall that d(G ) = 12 |G | when G is non-cyclic but contains a cyclic,
index 2 subgroup.
The Large Davenport Constant: New Results
Recall that d(G ) = 12 |G | when G is non-cyclic but contains a cyclic,
index 2 subgroup.
Theorem
Let G be a finite group with a cyclic, index 2 subgroup and let
G 0 = [G , G ] ≤ G be the commutator subgroup. Then
|G | − 1 if G is cyclic
D(G ) = d(G ) + |G 0 | and d(G ) =
1
if G is non-cyclic,
2 |G |
The Large Davenport Constant: Inductive Bounds
The Large Davenport Constant: Inductive Bounds
Theorem
Let G be a finite group and let H ≤ G be a subgroup. Then
D(G ) ≤ D(H)|G : H|.
The Large Davenport Constant: Inductive Bounds
Theorem
Let G be a finite group and let H ≤ G be a subgroup. Then
D(G ) ≤ D(H)|G : H|.
Theorem
Let G be a finite group with commutator subgroup G 0 = [G , G ] ≤ G and
let H E G be a normal subgroup such that G 0 ∩ H = {1}. Then
D(G ) ≤ D(H)D(G /H).
The Large Davenport Constant: The Group Fpq
The Large Davenport Constant: The Group Fpq
Recall that d(Fpq ) + 1 = p + q − 1, where Fpq is the non-abelian group
of order pq with p | q − 1 and p and q primes.
The Large Davenport Constant: The Group Fpq
Recall that d(Fpq ) + 1 = p + q − 1, where Fpq is the non-abelian group
of order pq with p | q − 1 and p and q primes.
Theorem
Let p and q be primes with p | q − 1. Then
D(Fpq ) = 2q.
The Large Davenport Constant: General Upper Bounds
The Large Davenport Constant: General Upper Bounds
Recall that, for G non-cyclic, we have d(G ) ≤ 12 |G |, and that
D(D2n ) = 2n = |G | for n odd.
The Large Davenport Constant: General Upper Bounds
Recall that, for G non-cyclic, we have d(G ) ≤ 12 |G |, and that
D(D2n ) = 2n = |G | for n odd.
Theorem
Let G be a finite, non-cyclic group and let p be the smallest prime divisor
of |G |. Then
2
D(G ) ≤ |G |.
p
The Large Davenport Constant: General Upper Bounds
Recall that, for G non-cyclic, we have d(G ) ≤ 12 |G |, and that
D(D2n ) = 2n = |G | for n odd.
Theorem
Let G be a finite, non-cyclic group and let p be the smallest prime divisor
of |G |. Then
2
D(G ) ≤ |G |.
p
When |G | is even, this is just the trivial bound D(G ) ≤ |G |, but in this
case we can do yet better.
The Large Davenport Constant: General Upper Bounds
Theorem
Let G be a finite, non-cyclic group and let p be the smallest prime divisor
of |G |. Then
2
D(G ) ≤ |G |.
p
Theorem
Let G be a finite group. Suppose G is neither cyclic nor dihedral of order
2n with n odd. Then
3
D(G ) ≤ |G |
4
Thanks to all!
A More Complicated Example
A More Complicated Example
Let
G = D2n = hα, τ : αn = 1 = τ 2 ,
be the Dihedral group of order 2n,
ατ = τ α−1 i
A More Complicated Example
Let
G = D2n = hα, τ : αn = 1 = τ 2 ,
ατ = τ α−1 i
be the Dihedral group of order 2n, and let
S = α[n−1] · τ · α[n−1] · τ,
where α[x] = α
. . · α}.
| · .{z
x
A More Complicated Example
Let
G = D2n = hα, τ : αn = 1 = τ 2 ,
ατ = τ α−1 i
be the Dihedral group of order 2n, and let
S = α[n−1] · τ · α[n−1] · τ,
where α[x] = α
. . · α}. Then
| · .{z
x
αn−1 τ αn−1 τ = α−1 τ α−1 τ = τ αα−1 τ = τ 2 = 1,
so S is a product-one sequence of length 2n = |G |.
A More Complicated Example
A More Complicated Example
Claim:
S = α[n−1] · τ · α[n−1] · τ
is an atom for n odd.
A More Complicated Example
Claim:
S = α[n−1] · τ · α[n−1] · τ
is an atom for n odd. Why?
A More Complicated Example
Claim:
S = α[n−1] · τ · α[n−1] · τ
is an atom for n odd. Why?
If
S = T1 · T2
with T1 and T2 product-one, then w.l.o.g.
T1 = α[n]
and
T2 = α[n−2] · τ [2] .
A More Complicated Example
Claim:
S = α[n−1] · τ · α[n−1] · τ
is an atom for n odd. Why?
If
S = T1 · T2
with T1 and T2 product-one, then w.l.o.g.
T1 = α[n]
and
T2 = α[n−2] · τ [2] .
But then
1 = αx τ αn−2−x τ = α2x−2 τ 2 = α2x−2
for some x ∈ [0, n − 2].
A More Complicated Example
Claim:
S = α[n−1] · τ · α[n−1] · τ
is an atom for n odd. Why?
If
S = T1 · T2
with T1 and T2 product-one, then w.l.o.g.
T1 = α[n]
and
T2 = α[n−2] · τ [2] .
But then
1 = αx τ αn−2−x τ = α2x−2 τ 2 = α2x−2
for some x ∈ [0, n − 2].
Hence 2x − 2 ≡ 0 mod n, implying (as n is odd) x − 1 ≡ 0 mod n,
which contradicts that x ∈ [0, n − 2].
A More Complicated Example
I
This shows that D(D2n ) = |D2n | = 2n for n odd.
A More Complicated Example
I
This shows that D(D2n ) = |D2n | = 2n for n odd.
I
Thus the upper bound D(G ) ≤ |G | is tight for Dihedral groups of
order 2n with n odd.
A More Complicated Example
I
This shows that D(D2n ) = |D2n | = 2n for n odd.
I
Thus the upper bound D(G ) ≤ |G | is tight for Dihedral groups of
order 2n with n odd.
I
So what can we say in general about upper bounds for D(G ) when
G is non-abelian?