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Groups, Graphs, and Symmetry: Cayley Graphs and theCayley Isomorphism Property1
Gregory Michel
Carleton College
November 3, 2013
1Joint work with Christopher Cox (Iowa State University) and Hannah Turner (BallState University) as a part of the 2013 REU at Iowa State University (NSF DMS0750986) under the guidance of Sung Y. Song (Iowa State University) and KathleenNowak (Iowa State University)
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
The Basics
Definition (Group)
A group G is a set that is closed under some binary associative operation ∗where
1 There is an identity element e for which a ∗ e = e ∗a = a for all a ∈ G .
2 Every element a ∈ G has an inverse a−1 for which aa−1 = a−1a = e
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Cayley Graphs
Definition (Cayley Graph)
Given a finite group G and a symmetric connector set S ⊆ G \ {e}, theCayley graph, denoted Cay(G ,S), is the graph with V = G andE = {(x , y) ∈ V × V : x−1y ∈ S} (i.e y = xs for some s ∈ S .)
Cay(Z9, {1, 3, 6, 8})
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Cayley Graphs
Definition (Cayley Graph)
Given a finite group G and a symmetric connector set S ⊆ G \ {e}, theCayley graph, denoted Cay(G ,S), is the graph with V = G andE = {(x , y) ∈ V × V : x−1y ∈ S} (i.e y = xs for some s ∈ S .)
Cay(Z9, {1, 3, 6, 8})
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Motivating Example: S3
Cay(S3, {(2 3), (1 2 3), (3 2 1)}) Cay(S3, {(1 2), (1 2 3), (3 2 1)})
Remark
These graphs are isomorphic!
Remark
If we let α be the inner automorphism defined by conjugating by theelement (1 3), then α({(2 3), (1 2 3), (3 2 1)}) = {(1 2), (1 2 3), (3 2 1)}.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Motivating Example: S3
Cay(S3, {(2 3), (1 2 3), (3 2 1)}) Cay(S3, {(1 2), (1 2 3), (3 2 1)})
Remark
These graphs are isomorphic!
Remark
If we let α be the inner automorphism defined by conjugating by theelement (1 3), then α({(2 3), (1 2 3), (3 2 1)}) = {(1 2), (1 2 3), (3 2 1)}.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Motivating Example: S3
Cay(S3, {(2 3), (1 2 3), (3 2 1)}) Cay(S3, {(1 2), (1 2 3), (3 2 1)})
Remark
These graphs are isomorphic!
Remark
If we let α be the inner automorphism defined by conjugating by theelement (1 3), then α({(2 3), (1 2 3), (3 2 1)}) = {(1 2), (1 2 3), (3 2 1)}.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Motivating Example: S4
Cay(S4, {(1 2)}) Cay(S4, {(1 2)(3 4)})
Remark
These graphs are isomorphic!
Remark
Automorphisms in S4 are all inner automorphisms, which preserve cyclestructure. Thus, there is no automorphism of S4 that sends (1 2) to(1 2)(3 4).
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Motivating Example: S4
Cay(S4, {(1 2)}) Cay(S4, {(1 2)(3 4)})
Remark
These graphs are isomorphic!
Remark
Automorphisms in S4 are all inner automorphisms, which preserve cyclestructure. Thus, there is no automorphism of S4 that sends (1 2) to(1 2)(3 4).
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Motivating Example: S4
Cay(S4, {(1 2)}) Cay(S4, {(1 2)(3 4)})
Remark
These graphs are isomorphic!
Remark
Automorphisms in S4 are all inner automorphisms, which preserve cyclestructure. Thus, there is no automorphism of S4 that sends (1 2) to(1 2)(3 4).
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
The Cayley Isomorphism Property
Definition (The Cayley-Isomorphism (CI) Property)
A Cayley graph of a group G with a symmetric subset S ⊆ G , Cay(G ,S),is said to be a CI-graph if, for any T such that Cay(G , S) ∼= Cay(G ,T ),there exists an α ∈ Aut(G ) such that α(S) = T . A group is said to be aCI-group if every Cayley graph of this group is a CI-graph.
Remark
Given a two sets S ,T ⊆ G for which α(S) = T for some automorphismα ∈ Aut(G ), Cay(G ,S) and Cay(G ,T ) will be graphically isomorphic.However, the converse does not necessarily hold.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
The Cayley Isomorphism Property
Definition (The Cayley-Isomorphism (CI) Property)
A Cayley graph of a group G with a symmetric subset S ⊆ G , Cay(G ,S),is said to be a CI-graph if, for any T such that Cay(G , S) ∼= Cay(G ,T ),there exists an α ∈ Aut(G ) such that α(S) = T . A group is said to be aCI-group if every Cayley graph of this group is a CI-graph.
Remark
Given a two sets S ,T ⊆ G for which α(S) = T for some automorphismα ∈ Aut(G ), Cay(G , S) and Cay(G ,T ) will be graphically isomorphic.However, the converse does not necessarily hold.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
The Cayley Isomorphism Property
Definition (The Cayley-Isomorphism (CI) Property)
A Cayley graph of a group G with a symmetric subset S ⊆ G , Cay(G ,S),is said to be a CI-graph if, for any T such that Cay(G , S) ∼= Cay(G ,T ),there exists an α ∈ Aut(G ) such that α(S) = T . A group is said to be aCI-group if every Cayley graph of this group is a CI-graph.
Remark
Given a two sets S ,T ⊆ G for which α(S) = T for some automorphismα ∈ Aut(G ), Cay(G , S) and Cay(G ,T ) will be graphically isomorphic.However, the converse does not necessarily hold.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Classifying Groups based on the CI-property
Claim
S3 is CI and S4 is not CI.
Conjecture (Adam’s Conjecture)
Zn is a CI group for all n.
Theorem (Muzychuk (2003))
The cyclic group Zn is a CI-group if and only if 8 - n, and p2 - n for anyodd prime p, or n ∈ {8, 9, 18}.
Question
Which groups are CI?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Classifying Groups based on the CI-property
Claim
S3 is CI and S4 is not CI.
Conjecture (Adam’s Conjecture)
Zn is a CI group for all n.
Theorem (Muzychuk (2003))
The cyclic group Zn is a CI-group if and only if 8 - n, and p2 - n for anyodd prime p, or n ∈ {8, 9, 18}.
Question
Which groups are CI?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Classifying Groups based on the CI-property
Claim
S3 is CI and S4 is not CI.
Conjecture (Adam’s Conjecture)
Zn is a CI group for all n.
Theorem (Muzychuk (2003))
The cyclic group Zn is a CI-group if and only if 8 - n, and p2 - n for anyodd prime p, or n ∈ {8, 9, 18}.
Question
Which groups are CI?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Classifying Groups based on the CI-property
Claim
S3 is CI and S4 is not CI.
Conjecture (Adam’s Conjecture)
Zn is a CI group for all n.
Theorem (Muzychuk (2003))
The cyclic group Zn is a CI-group if and only if 8 - n, and p2 - n for anyodd prime p, or n ∈ {8, 9, 18}.
Question
Which groups are CI?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Cayley Graphs and Subgroups
Claim
For a finite group G , let H = 〈S〉, H ≤ G . Γ = Cay(G ,S) has aconnected component Cay(H, S) with |H| vertices and is composed of[G : H] many disjoint isomorphic copies of this component.
Cay(S3, {(2 3), (1 2 3), (3 2 1)})
Cay(S4, {(2 3), (1 2 3), (3 2 1)})
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Cayley Graphs and Subgroups, cont.
Corollary
Cay(G ,S) is connected if and only if 〈S〉 = G .
Corollary
〈S〉 = S ∪ {e} if and only if each connected component is K|S|+1
Cay(S4, {(1 2), (3 4), (1 2)(3 4)}) ∼= 6 · K4
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Cayley Graphs and Subgroups, cont.
Corollary
Cay(G ,S) is connected if and only if 〈S〉 = G .
Corollary
〈S〉 = S ∪ {e} if and only if each connected component is K|S|+1
Cay(S4, {(1 2), (3 4), (1 2)(3 4)}) ∼= 6 · K4
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Subgroups of CI Groups
Theorem
Every subgroup of a CI-group is also CI.
Theorem
If G has a subgroup that is non-CI, then G is also non-CI.
Claim
Sn is non-CI for n ≥ 4.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Irreducibly Non-CI
Definition
A finite group G is called irreducibly non-CI (INCI) if G is a non-CI groupand every proper subgroup of G is CI.
{e}
Z2
Z2 × Z2 Z4
Z4 × Z2
{e}
Z2
Z3
Z2 × Z2
D4
Z4
S3
A4
S4
{e}Z2
Z4
Z3
Q8
Z6
SL(2, 3)
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Two Non-Isomorphic Subgroups of the Same Order
Theorem
If G has two non-isomorphic subgroups of the same order, then G isnon-CI.
Claim
Z4 × Z2 is non-CI.
Cay(Z4 × Z2,Z4 \ {e}) Cay(Z4 × Z2,Z2 × Z2 \ {e})
These graphs are isomorphic, and no automorphism of Z4 × Z2 will sendZ4 to Z2 × Z2.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
What’s Left?
{e}
Z2
Z2 × Z2 Z4
Z4 × Z2
{e}
Z2
Z3
Z2 × Z2
D4
Z4
S3
A4
S4
{e}Z2
Z4
Z3
Q8
Z6
SL(2, 3)
Theorem
G is non-CI if
1 If G has two non-isomorphic subgroups of the same order.
2 If G has a subgroup that is non-CI.
Claim
Cay(G ,S) is connected if and only if 〈S〉 = G .
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
What’s Left?
{e}
Z2
Z2 × Z2 Z4
Z4 × Z2
{e}
Z2
Z3
Z2 × Z2
D4
Z4
S3
A4
S4
{e}Z2
Z4
Z3
Q8
Z6
SL(2, 3)
Theorem
G is non-CI if
1 If G has two non-isomorphic subgroups of the same order.
2 If G has a subgroup that is non-CI.
Claim
Cay(G ,S) is connected if and only if 〈S〉 = G .
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Classification of Abelian Non-CI-Groups
Theorem
If an abelian group G is non-CI then one of the following two mutuallyexclusive conditions hold:
1 1 G has a proper subgroup that is non-CI (i.e. G is not INCI) AND/OR2 G has two non-isomorphic subgroups of the same order
2 All non-CI Cayley graphs of G are connected.
a0〈S〉a1〈S〉
a2〈S〉
b0〈T 〉b1〈T 〉
b2〈T 〉
a0
a1
a2
b0
b1
b2
× ×
γ(g) = γ(aks) = ϕ(ak)σ(α(s))
〈S〉 〈T 〉S α(S)
T
ϕ
α
σ
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Remaining Groups
Z16 and Z24
For Zn where 8|n and 8 6= nS = {1, 2, n2 − 1, n2 + 1, n − 2, n − 1} T = {1, n2 − 2, n2 − 1, n2 + 1, n2 + 2, n − 1}
SL(2, 3)
S =
{(0 12 2
),
(1 10 1
),
(0 21 2
),
(2 12 0
),
(1 20 1
),
(2 21 0
)}T =
{(1 21 0
),
(2 01 2
),
(1 12 0
),
(0 12 1
),
(2 02 2
),
(0 21 1
)}Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Open Problems
1 Extend the classification of subgroup group structures of INCI groupsto the non-abelian case (specifically Dedekind groups).
2 Why are INCI groups non-CI?
3 For what values of k is Zkp a CI group?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Open Problems
1 Extend the classification of subgroup group structures of INCI groupsto the non-abelian case (specifically Dedekind groups).
2 Why are INCI groups non-CI?
3 For what values of k is Zkp a CI group?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Open Problems
1 Extend the classification of subgroup group structures of INCI groupsto the non-abelian case (specifically Dedekind groups).
2 Why are INCI groups non-CI?
3 For what values of k is Zkp a CI group?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Open Problems
1 Extend the classification of subgroup group structures of INCI groupsto the non-abelian case (specifically Dedekind groups).
2 Why are INCI groups non-CI?
3 For what values of k is Zkp a CI group?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
References
M. Conder and Li C.H. “On isomorphisms for finite Cayley graphs”.In: European Journal of Combinatorics 19 (1998), pp. 911–919.
C.H. Li, Z.P. Lu, and Palfy P.P. “Further restriction on the structureof finite CI-groups”. In: Journal of Algebraic Combinatorics 26(2007), pp. 161–181.
M. Muzychuk. “Adam’s conjecture is true in the square-free case”.In: Journal of Combinatorial Theory 72 (1995), pp. 118–134.
M. Muzychuk. “An elementary abelian group of large rank is not aCI-group”. In: Discrete Mathematics 264 (2003), pp. 167–185.
M. Muzychuk. “On Adam’s conjecture for circulant graphs”. In:Discrete Mathematics 167 (1997), pp. 495–510.
M.Y. Xu. “Automorphism groups and isomorphisms of Cayleydigraphs”. In: Discrete Mathematics 182 (1998), pp. 309–320.
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Acknowledgments
Thank you to Eric Egge, Adam Berliner, St. Olaf College, and the theNUMS Conference.
Thank you to my mentors Dr. Sung Y. Song and Kathleen Nowakand to my co-researchers Christopher Cox and Hannah Turner.
Thank you to the Iowa State University for hosting this research andthe NSF (NSF DMS 0750986) for funding.
Thank you all for coming!
Questions?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Acknowledgments
Thank you to Eric Egge, Adam Berliner, St. Olaf College, and the theNUMS Conference.
Thank you to my mentors Dr. Sung Y. Song and Kathleen Nowakand to my co-researchers Christopher Cox and Hannah Turner.
Thank you to the Iowa State University for hosting this research andthe NSF (NSF DMS 0750986) for funding.
Thank you all for coming!
Questions?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Acknowledgments
Thank you to Eric Egge, Adam Berliner, St. Olaf College, and the theNUMS Conference.
Thank you to my mentors Dr. Sung Y. Song and Kathleen Nowakand to my co-researchers Christopher Cox and Hannah Turner.
Thank you to the Iowa State University for hosting this research andthe NSF (NSF DMS 0750986) for funding.
Thank you all for coming!
Questions?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
Acknowledgments
Thank you to Eric Egge, Adam Berliner, St. Olaf College, and the theNUMS Conference.
Thank you to my mentors Dr. Sung Y. Song and Kathleen Nowakand to my co-researchers Christopher Cox and Hannah Turner.
Thank you to the Iowa State University for hosting this research andthe NSF (NSF DMS 0750986) for funding.
Thank you all for coming!
Questions?
Gregory Michel Algebraic Graph Theory (NSF DMS 0750986) November 3, 2013
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