The
Symmetric
Group:
The
Symmetric
Group:
Representations
Combinatorics
Representations
andand
Combinatorics
Royal Holloway,
of Londonof a
Holloway,
University
of London
TheRoyal
probability
that University
two
elements
finite group are
conjugate
Speakers:
29 March
Speakers:
29 March
2011 2011
Christine
Bessenrodt
(Hannover)
Forand
more
information
Christine
Bessenrodt
(Hannover)
For more
information
see see
Mark
Wildon
(with
Simon
R. Blackburn
John
R. Britnell)
www.ma.rhul.ac.uk/∼uvah099
www.ma.rhul.ac.uk/∼uvah099
Matthew
(QMUL)
Matthew
FayersFayers
(QMUL)
Supported
an LMS
Scheme 1 grant
Supported by
an LMS by
Scheme
1 grant
Mark Wildon
(RHUL)
Mark Wildon
(RHUL)
Outline
§1 Small κ(G )
§2 Limit points of κ(G )
§3 Large κ(G )
§4 κ for symmetric groups
Inductive proof of Theorem 3
5.6
n2 κ(Sn )
upper bound for n2 κ(Sn )
optimal k (right axis)
5.4
25
5
4.8
20
4.6
4.4
15
4.2
4
20
40
60
80
100
n
120
140
160
180
10
200
optimal k
n2 κ(Sn )
5.2
30
Conjugate commuting probabilities
Let ρ(G ) be the probability that if g , h ∈ G are chosen uniformly
and independently at random, then g and h have conjugates that
commute.
The method used to prove Theorem 3 also give the analogous
result for ρ(Sn ).
Theorem 4
For all n ∈ N we have
ρ(Sn ) ≤
Cρ
n2
where Cρ = 102 ρ(S10 ) ≈ 11.42. Moreover if Aρ =
A
then ρ(Sn ) ∼ n2ρ as n → ∞.
P∞
n=1 ρ(Sn )
Conjugate commuting probabilities
Let ρ(G ) be the probability that if g , h ∈ G are chosen uniformly
and independently at random, then g and h have conjugates that
commute.
The method used to prove Theorem 3 also give the analogous
result for ρ(Sn ).
Theorem 4
For all n ∈ N we have
ρ(Sn ) ≤
Cρ
n2
where Cρ = 102 ρ(S10 ) ≈ 11.42. Moreover if Aρ =
A
then ρ(Sn ) ∼ n2ρ as n → ∞.
P∞
n=1 ρ(Sn )
Corollary 5
Let g , h ∈ Sn be chosen independently and uniformly at random.
If g and h have conjugates that commute then, provided n is
sufficiently large, the probability that g and h are conjugate is at
least 42/65 ≈ 0.646, and at most 43/61 ≈ 0.710.
Thank you. Any questions?
Other results and questions
I
If G and H are isoclinic groups then κ(G ) = κ(H). The
groups G such that |G : Z (G )| = 4 form a single isoclinism
class.
I
If N C G then κ(G ) < κ(G /N). Using this we can define κ
for a profinite group G as limN κ(G /N). Could it be that
every limit point of κ is ’explained’ by a profinite group?
I
What can be said about ρ(G ) for a general group G ?