Upper and Lower Semimodularity of the Supercharacter Theory Lattices - - PowerPoint PPT Presentation

Upper and Lower Semimodularity of the Supercharacter Theory Lattices of Cyclic Groups

Samuel Benidt, William Hall, & Anders Hendrickson November 10, 2009

Definitions Our Work Two main theorems

Outline

1

Definitions Lattices Groups and subgroups Supercharacter Theories

2

Our Work Goals and Strategy Analysis of Specific Cases

3

Two main theorems Upper Semimodularity Lower Semimodularity

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Lattices

As you may know, a partially ordered set is a set with an order that requires the following for elements a, b in the set: either a ≤ b

• r a ≥ b
• r a and b are incomparable

Definition A lattice is a partially ordered set in which any two elements a and b have a unique least upper bound, a ∨ b, and greatest lower bound, a ∧ b.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Examples

A Lattice Definition We say an element a of a lattice covers another element d if a ≥ d and there are no elements between a and d. We write d a.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Examples

A Lattice Not a lattice Definition We say an element a of a lattice covers another element d if a ≥ d and there are no elements between a and d. We write d a.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Examples

A Lattice Not a lattice Definition We say an element a of a lattice covers another element d if a ≥ d and there are no elements between a and d. We write d a.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Upper Semimodularity

Definition A lattice L of finite length, is said to be upper semimodular if the following condition is satisfied: if a ∧ b a, b then a, b a ∨ b.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Upper Semimodularity

Definition A lattice L of finite length, is said to be upper semimodular if the following condition is satisfied: if a ∧ b a, b then a, b a ∨ b. ⇒

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Lower Semimodularity

Definition Let L be a lattice of finite length. Then L is lower semimodular if for all a, b ∈ L, if a, b a ∨ b then a ∧ b a, b

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Lower Semimodularity

Definition Let L be a lattice of finite length. Then L is lower semimodular if for all a, b ∈ L, if a, b a ∨ b then a ∧ b a, b ⇒

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Group

Definition A set G, with respect to operation ∗, is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Group

Definition A set G, with respect to operation ∗, is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Group

Definition A set G, with respect to operation ∗, is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Group

Definition A set G, with respect to operation ∗, is a group if it is associative has an identity (equivalently, contains 1) is closed has inverses We will only consider cyclic groups, in which there is a generator g ∈ G such that each element in the group equals g n for some n ∈ N.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Subgroup

Definition A subset of a group is a subgroup if it is a group in its own right. Subgroups of Z12 Group Elements Z12 1, g, g 2, g 3, g 4, g 5, g 6, g 7, g 8, g 9, g 10, g 11 Z6 1, g 2, g 4, g 6, g 8, g 10 Z4 1, g 3, g 6, g 9 Z3 1, g 4, g 8 Z2 1, g 6 Z1 1

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Supercharacter Theory

Definition Let G be a finite group and let K be a partition of G. Then we say K is a superchacter theory if the following three conditions hold: {1} is a part of K. A product of parts of K is a linear combination of the parts of K. Each part of K is a union of some conjugacy classes of G. Define Sup(G) as the set of all supercharacter theories of the group G.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Sup(G) as a Lattice

Note that Sup(G) is indeed a poset and can thus be considered in terms

• f lattice theory.

a1 ☞ ☞ ☞a2, a3 ☞ ☞ ☞a4 a1, a2 ☞ ☞ ☞a3 ☞ ☞ ☞a4 X Y

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Sup(G) as a Lattice

Note that Sup(G) is indeed a poset and can thus be considered in terms

• f lattice theory.

a1 ☞ ☞ ☞a2, a3 ☞ ☞ ☞a4 a1, a2 ☞ ☞ ☞a3 ☞ ☞ ☞a4 a1, a2, a3 ☞ ☞ ☞a4 X ∨ Y X Y

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Sup(G) as a Lattice

Note that Sup(G) is indeed a poset and can thus be considered in terms

• f lattice theory.

a1 ☞ ☞ ☞a2, a3 ☞ ☞ ☞a4 a1, a2 ☞ ☞ ☞a3 ☞ ☞ ☞a4 a1, a2, a3 ☞ ☞ ☞a4 a1 ☞ ☞ ☞a2 ☞ ☞ ☞a3 ☞ ☞ ☞a4 X ∧ Y X ∨ Y X Y

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Some Examples: Sup(Z3)

Figure: The graph of Sup(Z3)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Some Examples: Sup(Z17)

Figure: The graph of Sup(Z17)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Some Examples: Sup(Z51)

Figure: The graph of Sup(Z51)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

Finding Supercharacter Theories

The following are some methods of discovering supercharacter theories that will be referenced shortly: minimal supercharacter theory: m(Zn) maximal supercharacter theory: M(Zn) inverse supercharacter theory: Inv(Zn) = {1/g, g −1/g 2, g −2/...} partition the elements by their orders: A(Zn) ∗-product supercharacter theory: X ∗ Y , where X ∈ Sup(N) and Y ∈ Sup(G/N).

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

∗-product

As mentioned on the previous slide, one way to find supercharacter theories is a method called the ∗-product. Let N be normal in G with X ∈ Sup(N), Y ∈ Sup(G/N). Then X ∗ Y ∈ Sup(G). X Y

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

∗-product

As mentioned on the previous slide, one way to find supercharacter theories is a method called the ∗-product. Let N be normal in G with X ∈ Sup(N), Y ∈ Sup(G/N). Then X ∗ Y ∈ Sup(G). X Y

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Lattices Groups and subgroups Supercharacter Theories

∗-product

As mentioned on the previous slide, one way to find supercharacter theories is a method called the ∗-product. Let N be normal in G with X ∈ Sup(N), Y ∈ Sup(G/N). Then X ∗ Y ∈ Sup(G). X Y X ∗ Y

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Research Goal

Research Goal We seek necessary and sufficient conditions for when Sup(G) is upper- and lower-semimodular.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Inheritance Lemma

Lemma (Inheritance) Let N be a subgroup of cyclic group G. Then

1

If Sup(N) is not upper semimodular, then Sup(G) is not upper semimodular.

2

If Sup(N) is not lower semimodular, then Sup(G) is not lower semimodular.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p USM LSM

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = p

Lemma Given a group G of order p, Sup(G) is isomorphic to the subgroup lattice

• f the automorphism group of G.

Lemma It is a well-known fact in lattice theory that the subgroup lattice of an Abelian group is both USM and LSM. Therefore, Sup(G) is both USM and LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = p

Lemma Given a group G of order p, Sup(G) is isomorphic to the subgroup lattice

• f the automorphism group of G.

Lemma It is a well-known fact in lattice theory that the subgroup lattice of an Abelian group is both USM and LSM. Therefore, Sup(G) is both USM and LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = p

Lemma Given a group G of order p, Sup(G) is isomorphic to the subgroup lattice

• f the automorphism group of G.

Lemma It is a well-known fact in lattice theory that the subgroup lattice of an Abelian group is both USM and LSM. Therefore, Sup(G) is both USM and LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Example of |G| = p

[1] [18] [7] [8] [4] [2]

Lattice of Sup(Z19) Lattice of subgroups

• f Aut(Z19)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p USM LSM

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p USM Yes LSM Yes

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 USM Yes LSM Yes

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = p2, if p is odd

If G is cyclic of order p2, p an odd prime, then Sup(G) contains the following sublattice.

m(Zp2) m(Zp) ∗ m(Zp) Inv(Zp2) Inv(Zp) ∗ m(Zp) m(Zp) ∗ Inv(Zp) Inv(Zp) ∗ Inv(Zp)

Note that each line drawn in on this particular diagram indicates a covering relation, and each element listed is distinct from the others.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = p2, if p is odd

If G is cyclic of order p2, p an odd prime, then Sup(G) contains the following sublattice.

m(Zp2) m(Zp) ∗ m(Zp) Inv(Zp2) Inv(Zp) ∗ m(Zp) m(Zp) ∗ Inv(Zp) Inv(Zp) ∗ Inv(Zp)

Note that each line drawn in on this particular diagram indicates a covering relation, and each element listed is distinct from the others.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = p2, if p is odd

If G is cyclic of order p2, p an odd prime, then Sup(G) contains the following sublattice.

m(Zp2) m(Zp) ∗ m(Zp) Inv(Zp2) Inv(Zp) ∗ m(Zp) m(Zp) ∗ Inv(Zp) Inv(Zp) ∗ Inv(Zp)

Note that each line drawn in on this particular diagram indicates a covering relation, and each element listed is distinct from the others.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = 4

m(Z4) M(Z2) ∗ M(Z2) M(Z4)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 USM Yes LSM Yes

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 4 USM Yes No∗ Yes LSM Yes No∗ Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq 4 USM Yes No∗ Yes LSM Yes No∗ Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

USM Case for |G| = pq

m(Zpq) m(Zp) ∗ m(Zq) m(Zq) ∗ m(Zp) M(Zp) ∗ M(Zq) M(Zpq)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

USM Case for |G| = pq

m(Zpq) m(Zp) ∗ m(Zq) m(Zq) ∗ m(Zp) M(Zp) ∗ M(Zq) M(Zpq)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq 4 USM Yes No∗ Yes LSM Yes No∗ Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq 4 USM Yes No∗ No Yes LSM Yes No∗ Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

LSM Case for |G| = pq

Lemma Let G be a cyclic group of order pq. Then Sup(G) is lower semimodular.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq 4 USM Yes No∗ No Yes LSM Yes No∗ Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq 4 USM Yes No∗ No Yes LSM Yes No∗ Yes Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq pqr 4p 4 USM Yes No∗ No Yes LSM Yes No∗ Yes Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = pqr

Aut(Zpq) ∗ M(Zr) M(Zp) ∗ M(Zq) ∗ M(Zr) M(Zq) ∗ M(Zpr) M(Zp) ∗ M(Zqr) M(Zpqr)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = pqr

Aut(Zpq) ∗ M(Zr) M(Zp) ∗ M(Zq) ∗ M(Zr) M(Zq) ∗ M(Zpr) M(Zp) ∗ M(Zqr) M(Zpqr)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = 4p, if p is odd

A(Z2p) ∗ M(Z2) M(Z2) ∗ M(Zp) ∗ M(Z2) M(Zp) ∗ M(Z2) ∗ M(Z2) M(Z2) ∗ M(Z2p) M(Zp) ∗ M(Z4) M(Z4p)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = 4p, if p is odd

A(Z2p) ∗ M(Z2) M(Z2) ∗ M(Zp) ∗ M(Z2) M(Zp) ∗ M(Z2) ∗ M(Z2) M(Z2) ∗ M(Z2p) M(Zp) ∗ M(Z4) M(Z4p)

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = 4p, if p is even

G ∼ = Z8

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Case where |G| = 4p, if p is even

G ∼ = Z8

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Goals and Strategy Analysis of Specific Cases

Summary Table Summary Table

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

∗for odd primes p

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

Upper Semimodularity Theorem

Theorem (USM) Let G be a cyclic group. Then Sup(G) is upper semimodular if and only if the order of G is prime or four.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is USM: If |G| = p, then we proved that Sup(G) is USM. If |G| = 4, we verified by picture that this supercharacter theory lattice is USM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is USM: If |G| = p, then we proved that Sup(G) is USM. If |G| = 4, we verified by picture that this supercharacter theory lattice is USM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is USM: If |G| = p, then we proved that Sup(G) is USM. If |G| = 4, we verified by picture that this supercharacter theory lattice is USM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is USM: If |G| = p, then we proved that Sup(G) is USM. If |G| = 4, we verified by picture that this supercharacter theory lattice is USM. What happens for other orders of G?

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

If |G| is not prime or four, then it is either a multiple of two distinct primes or a prime power. For |G| = k · pq, where p = q, then we showed that Sup(Zpq) is not

• USM. By the Inheritance Lemma, Sup(G) is not USM either.

For |G| = pn, where p = 2 and n ≥ 2, we again showed that Sup(Zp2) is not USM. Then, by the Inheritance Lemma, Sup(G) is not USM. This leaves |G| = 2n with n ≥ 3. We verified that Sup(Z8) is not USM, and thus, for all higher 2-powers, Sup(G) is not USM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

If |G| is not prime or four, then it is either a multiple of two distinct primes or a prime power. For |G| = k · pq, where p = q, then we showed that Sup(Zpq) is not

• USM. By the Inheritance Lemma, Sup(G) is not USM either.

For |G| = pn, where p = 2 and n ≥ 2, we again showed that Sup(Zp2) is not USM. Then, by the Inheritance Lemma, Sup(G) is not USM. This leaves |G| = 2n with n ≥ 3. We verified that Sup(Z8) is not USM, and thus, for all higher 2-powers, Sup(G) is not USM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

If |G| is not prime or four, then it is either a multiple of two distinct primes or a prime power. For |G| = k · pq, where p = q, then we showed that Sup(Zpq) is not

• USM. By the Inheritance Lemma, Sup(G) is not USM either.

For |G| = pn, where p = 2 and n ≥ 2, we again showed that Sup(Zp2) is not USM. Then, by the Inheritance Lemma, Sup(G) is not USM. This leaves |G| = 2n with n ≥ 3. We verified that Sup(Z8) is not USM, and thus, for all higher 2-powers, Sup(G) is not USM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

If |G| is not prime or four, then it is either a multiple of two distinct primes or a prime power. For |G| = k · pq, where p = q, then we showed that Sup(Zpq) is not

• USM. By the Inheritance Lemma, Sup(G) is not USM either.

For |G| = pn, where p = 2 and n ≥ 2, we again showed that Sup(Zp2) is not USM. Then, by the Inheritance Lemma, Sup(G) is not USM. This leaves |G| = 2n with n ≥ 3. We verified that Sup(Z8) is not USM, and thus, for all higher 2-powers, Sup(G) is not USM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

USM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

If |G| is not prime or four, then it is either a multiple of two distinct primes or a prime power. For |G| = k · pq, where p = q, then we showed that Sup(Zpq) is not

• USM. By the Inheritance Lemma, Sup(G) is not USM either.

For |G| = pn, where p = 2 and n ≥ 2, we again showed that Sup(Zp2) is not USM. Then, by the Inheritance Lemma, Sup(G) is not USM. This leaves |G| = 2n with n ≥ 3. We verified that Sup(Z8) is not USM, and thus, for all higher 2-powers, Sup(G) is not USM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

Lower Semimodularity Theorem

Theorem (LSM) Let G be a cyclic group. Then Sup(G) is lower semimodular if and only if the order of G is prime, the product of two distinct primes, or four.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is LSM: When |G| = p, Sup(G) is lattice-isomorphic to the lattice C of subgroups of Aut(G). Since C is LSM, Sup(G) is also. When |G| = pq, where p = q, a previously-stated lemma proved that Sup(G) is LSM. When |G| = 4, we verified by picture that this supercharacter theory lattice is LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is LSM: When |G| = p, Sup(G) is lattice-isomorphic to the lattice C of subgroups of Aut(G). Since C is LSM, Sup(G) is also. When |G| = pq, where p = q, a previously-stated lemma proved that Sup(G) is LSM. When |G| = 4, we verified by picture that this supercharacter theory lattice is LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is LSM: When |G| = p, Sup(G) is lattice-isomorphic to the lattice C of subgroups of Aut(G). Since C is LSM, Sup(G) is also. When |G| = pq, where p = q, a previously-stated lemma proved that Sup(G) is LSM. When |G| = 4, we verified by picture that this supercharacter theory lattice is LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is LSM: When |G| = p, Sup(G) is lattice-isomorphic to the lattice C of subgroups of Aut(G). Since C is LSM, Sup(G) is also. When |G| = pq, where p = q, a previously-stated lemma proved that Sup(G) is LSM. When |G| = 4, we verified by picture that this supercharacter theory lattice is LSM.

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Cases where cyclic group G is LSM: When |G| = p, Sup(G) is lattice-isomorphic to the lattice C of subgroups of Aut(G). Since C is LSM, Sup(G) is also. When |G| = pq, where p = q, a previously-stated lemma proved that Sup(G) is LSM. When |G| = 4, we verified by picture that this supercharacter theory lattice is LSM. What happens for other orders of G?

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Definitions Our Work Two main theorems Upper Semimodularity Lower Semimodularity

LSM Proof cont.

p p2 pq pqr 4p 4 USM Yes No∗ No No No Yes LSM Yes No∗ Yes No No Yes

Suppose |G| is not p, pq, or 4. Consider the number of distinct prime factors of |G|. 1: If |G| = pa, then

If p is odd, then p2☞

☞|G|. So Sup(G) is not LSM.

If p is even, then 4p = 8

☞ ☞|G|, so Sup(G) is not LSM. 2: If |G| = paqb, then wlog a ≥ 2.

If p is odd, then p2☞

☞|G|, so Sup(G) is not LSM.

If p = 2, then 4q

☞ ☞|G| so Sup(G) is not LSM. ≥ 3: If |G| has at least 3 distinct prime factors, then pqr ☞ ☞|G|, so Sup(G) is not LSM. ✷

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

Questions?

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups