Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
The Conjugacy Probability
Misja F.A. Steinmetz
University of Cambridge
8 December 2014
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
1 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
References
Main references:
Blackburn, Britnell and Wildon - The Probability
that Group Elements are Conjugate [2012]
S. and Whybrow - The Asymptotic Behaviour of the
Conjugacy Probability of the Alternating Group
[preprint]
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
3 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
4 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Background
Suppose G is a finite group (assumed throughout talk).
How do we measure how far away G is from being abelian?
Definition (The Commuting Probability)
cp(G ) := the probability that two elements from the group G
commute, when chosen uniformly and independently at random.
• First defined in GUSTAFSON [1973].
• This is a quite well-studied statistic.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Conjugacy Probability
Recall: g , h ∈ G are conjugate if there exists an x ∈ G such that
h = g x := x −1 gx.
By contrast, arising just as naturally as cp(G ), we get the statistic:
Definition (The Conjugacy Probability)
κ(G ) := the probability that two elements from the group G are
conjugate, when chosen uniformly and independently at random.
• First defined in BLACKBURN, BRITNELL & WILDON [2012].
• This is a very little studied statistic.
• The theory has many points of similarity with the theory of
cp(G ).
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Side Note
As an irrelevant side note, BBW also define and study the
following natural statistic:
Definition
ρ(G ) := the probability that two elements from the group G have
conjugates that commute, when chosen uniformly and
independently at random.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
The First Question on Everyone’s Mind
Back to the conjugacy probability...
What is the conjugacy probability of the monster group M?
κ(M) =
3988143668287803887642853139272815429573859920421242687028866068819549244425816512406261807945759882702801
326446079385778135624234403861132965065140757627538960257020134575151234105353132375539712000000000000000000
≈ 0.0122168527.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
9 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Elementary Identities
Let k(G ) be the number of conjugacy classes of G . In
GUSTAFSON [1973] we find the elementary identity
cp(G ) =
k(G )
.
|G |
(Actually derives from ERDÖS-TURÁN [1968], but not stated in
terms of probability there.)
Proof.
k(G )
1 X 1
1 X |CentG (g )|
=
=
|G |
|G |
|G |
|G |
|g G |
g ∈G
g ∈G
1
=
|{(g , h) ∈ G × G : gh = hg }| = cp(G ).
|G |2
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
The Conjugacy Probability Identity
Suppose g1 , . . . , gk is a complete set of representatives of the
conjugacy classes of G . Then
κ(G ) = the probability that a pair of elements is conjugate
|{(g , h) ∈ G × G : g and h are conjugate}|
=
.
|G |2
Hence,
κ(G ) =
k
X
|g G |2
i
i=1
Misja F.A. Steinmetz
(University of Cambridge)
|G |2
=
k
X
i=1
1
|CentG (gi )|2
The Conjugacy Probability
.
8 December 2014
11 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Gap results
Note that G is abelian ⇐⇒ cp(G ) = 1.
Fact (Gustafson)
If cp(G ) 6= 1, then cp(G ) ≤
(e.g. D8 ).
5
8
with equality iff |G : Z (G )| = 4
Proof.
Every non-central class has size at least 2. So
k(G )≤ |Z (G )| + 12 (|G | − |Z (G )|). Hence
cp(G ) ≤
1
1
+
,
2 2|G : Z (G )|
and, since G /Z (G ) cannot be cyclic, it has order ≥ 4. When
G /Z (G ) ∼
= C2 × C2 , every non-central class has size 2, so the
bound is sharp.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
12 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Gap results (ctd.)
Compare this with
Fact (BBW)
If κ(G ) 6= |G1 | , then κ(G ) ≥
|G : Z (G )| = 4 (e.g. D8 ).
7
4|G |
with equality if and only if
(Note that G is abelian ⇐⇒ κ(G ) =
1
|G | .)
So both of these facts are ’gap’ results between the values of the
statistics on abelian and non-abelian groups and the extremal
examples in each case are the same.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Isoclinic Groups
Definition
Two groups G1 and G2 are isoclinic if there exist isomorphisms
α:
G1
G2
→
and β : G10 → G20
Z (G1 )
Z (G2 )
which respect the commutator map
[, ]:
G
G
×
→ G 0,
Z (G ) Z (G )
i.e. for x, y ∈ G1 , we have that
[α(xZ (G1 )), α(yZ (G1 ))] = β([xZ (G1 ), yZ (G1 )]).
(Introduced by HALL [1940] in the context of p-groups, but
concept arises naturally elsewhere too.)
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
14 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Stability on Isoclinism Classes
Fact (LESCOT [1995])
If G1 and G2 are isoclinic groups, then cp(G1 ) = cp(G2 ).
Compare with
Fact (BBW)
If G1 and G2 are isoclinic groups, then |G1 |κ(G1 ) = |G2 |κ(G2 ).
This should convince you that the theories of cp(G ) and κ(G )
have important similarities.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
15 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
16 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Classification of G with large κ(G )
The first main result of BBW: can classify all groups with very
large values of κ(G ).
Theorem (BBW)
Let G be a non-trivial finite group. Then κ(G ) ≥
one of the following holds:
1
4
if and only if
(i) |G | ≤ 4;
(ii) G ∼
= A4 , S4 , A5 ;
(iii) G ∼
= C7 o C3 :=< g , h : g 7 = h3 = 1, g h = g 2 >;
(iv) G ∼
= A o C2 where A is an abelian group of odd order and C2
acts on A by inversion.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
17 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Second Gap Result
We get a second easy gap result of κ(G ) for free:
Corollary (BBW)
If G is a finite group such that κ(G ) > 21 , then κ(G ) = 1, i.e. G is
trivial.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
19 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Conjugacy Probability of Sn
We now turn our attention to the more specific case of G = Sn .
Recall: permutations in Sn are conjugate if and only if they have
the same cycle type.
Lemma
Let n be a positive integer. Then
κ(Sn ) ≥
1
.
n2
Proof.
We have (n − 1)! n-cycles in Sn , which are all conjugate. So the
n-cycles contribute n12 to κ(Sn ).
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Conjugacy Probability of Sn
At least a little surprising that we get the following picture when
we plot n2 κ(Sn ):
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Asymptotic Behaviour of κ(Sn )
This serves as numerical evidence for two theorems:
Theorem (BBW)
For all positive integers n we have κ(Sn ) ≤ Cκ /n2 , where
Cκ = 132 κ(S13 ).
Theorem P
Let Aκ =
∞
n=1 κ(Sn )
≈ 4.26340. Then
lim n2 κ(Sn ) = Aκ .
n→∞
Second theorem was first proved by FLAJOLET, FUSY,
GOURDON, PARANARIO and POUYANNE [2006] in terms of
partitions using advanced methods in analytic combinatorics.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Heuristic Argument of Proof
Fact
If k > 12 n, then the probability that σ ∈ Sn contains a k-cycle is k1 .
Fact
The conjugacy probability of Sn is almost completely determined
by elements in Sn that contain a large cycle, i.e. a cycle of length
n − k for k << n.
Now we see that for large n
1
κ(Sk )
(n − k)2
k<<n
1 X
κ(Sk )
≈ 2
n
κ(Sn ) ≈
X
k<<n
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
24 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Numerical Evidence
Does something similar happen for An ?
We plot n2 κ(An ):
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Split Conjugacy Classes
Fact (Splitting Criterion)
Let g ∈ An . Then
(
1 Sn
|g | if g only has cycles of distinct odd lengths;
An
|g | = 2 S
|g n |
otherwise.
Definition (SW)
Suppose σ, τ ∈ Sn chosen independently and uniformly at random.
• κE (Sn ) := probability that σ, τ are conjugate, given that they
are even permutations;
• κO (Sn ) := probability that σ, τ are conjugate, given that they
are odd permutations;
• Q(Sn ) := probability that σ, τ have the same cycle type and
are only composed of cycles of distinct odd lengths.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Towards Asymptotic Theorem
Theorem (SW)
There exist real constants C0 , C1 and C2 such that for all n ∈ Z>0
κE (Sn ) ≤
C0
C1
C2
, κO (Sn ) ≤ 2 and Q(Sn ) ≤ 2 .
2
n
n
n
Lemma (SW)
For any n ∈ Z>0 we have the equality
κ(An ) = κE (Sn ) − 2Q(Sn ).
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Asymptotic Theorem for κ(An )
Theorem (SW)
Using notation as above, we have
lim n2 κ(An ) =
n→∞
n even
∞
X
κO (Sd ) +
d even
∞
X
(κE (Sd ) − 2Q(Sd ))
d odd
and
2
lim n κ(An ) =
n→∞
n odd
Misja F.A. Steinmetz
∞
X
(κE (Sd ) − 2Q(Sd )) +
d even
(University of Cambridge)
∞
X
κO (Sd ).
d odd
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Table of Contents
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
29 / 32
Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Related Open Problem 1
Let q(Sn ) be the probability that a single permutation in Sn only
has cycles of distinct odd lengths. Do we see similar asymptotic
√
behaviour? Strong numerical evidence suggests q(Sn ) ∼ C / n as
n → ∞.
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Related Open Problem 2
Does κ(GLd (q)) satisfy a similar asymptotic identity? Intuitively
one would expect that κ(GLd (q)) ∼ C /q d as d → ∞. Some weak
numerical evidence available for q = 2, 3 which neither contradicts
nor confirms this suspicion. Computations get very large very
quickly.
Figure: graph of 2d GLd (2)
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
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Introduction
Similarities between cp and κ
Classification of G with large κ(G )
Sn case
An case
Related Problems
Thank you for you attention.
Questions?
Misja F.A. Steinmetz
(University of Cambridge)
The Conjugacy Probability
8 December 2014
32 / 32