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When is the lattice of closure operators on a subgroup lattice again - - PowerPoint PPT Presentation

Jun 10, 2023 654 likes •1.43k views

When is the lattice of closure operators on a subgroup lattice again a subgroup lattice? Martha Kilpack 1 and Arturo Magidin 2 1 Brigham Young University 2 University of Louisiana at Lafayette Groups St Andrews 2017 Martha Kilpack and Arturo

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When is the lattice of closure operators on a subgroup lattice again a subgroup lattice?

Martha Kilpack1 and Arturo Magidin2

1Brigham Young University 2University of Louisiana at Lafayette

Groups St Andrews 2017

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Lattices

Definition A partially ordered set L is a lattice if any two elements x, y ∈ L have a least upper bound x ∨ y and a greatest lower bound x ∧ y.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Lattices

Definition A partially ordered set L is a lattice if any two elements x, y ∈ L have a least upper bound x ∨ y and a greatest lower bound x ∧ y. Typical examples include the lattice of subsets of a set, or the lattice of subalgebras of an algebra (in the sense of universal algebra), ordered by inclusion.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Complete lattices

A lattice L is complete if any subset S ⊆ L has a least upper bound and a greatest lower bound.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Complete lattices

A lattice L is complete if any subset S ⊆ L has a least upper bound and a greatest lower bound. Birkhoff proved that every complete lattice L is the lattice of subalgebras of some (possibly infinitary) algebra.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Complete lattices

A lattice L is complete if any subset S ⊆ L has a least upper bound and a greatest lower bound. Birkhoff proved that every complete lattice L is the lattice of subalgebras of some (possibly infinitary) algebra. Whitman (1946) proved that every lattice is isomorphic to a sublattice of the lattice of subgroups of some group.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Complete lattices

A lattice L is complete if any subset S ⊆ L has a least upper bound and a greatest lower bound. Birkhoff proved that every complete lattice L is the lattice of subalgebras of some (possibly infinitary) algebra. Whitman (1946) proved that every lattice is isomorphic to a sublattice of the lattice of subgroups of some group. Túma (1989) proved that every algebraic lattice is isomorphic to an interval in the lattice of subgroups of some (possibly infinite) group.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Complete lattices

A lattice L is complete if any subset S ⊆ L has a least upper bound and a greatest lower bound. Birkhoff proved that every complete lattice L is the lattice of subalgebras of some (possibly infinitary) algebra. Whitman (1946) proved that every lattice is isomorphic to a sublattice of the lattice of subgroups of some group. Túma (1989) proved that every algebraic lattice is isomorphic to an interval in the lattice of subgroups of some (possibly infinite) group. Question Given a class of lattices, which ones are isomorphic to the lattice of subgroups of a group K?

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators

Definition Let P be a partially ordered set. A closure operator on P is a function cl: P → P such that for all x, y ∈ P:

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators

Definition Let P be a partially ordered set. A closure operator on P is a function cl: P → P such that for all x, y ∈ P: (i) x ≤ cl(x) (increasing or extensive);

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators

Definition Let P be a partially ordered set. A closure operator on P is a function cl: P → P such that for all x, y ∈ P: (i) x ≤ cl(x) (increasing or extensive); (ii) If x ≤ y then cl(x) ≤ cl(y) (isotone);

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators

Definition Let P be a partially ordered set. A closure operator on P is a function cl: P → P such that for all x, y ∈ P: (i) x ≤ cl(x) (increasing or extensive); (ii) If x ≤ y then cl(x) ≤ cl(y) (isotone); (iii) cl(cl(x)) = cl(x) (idempotent).

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Comparing closure operators

If P is a partially ordered set, and φ, ψ are closure operators on P, we say φ ψ if and only if φ(x) ≤ ψ(x) for all x ∈ P.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Comparing closure operators

If P is a partially ordered set, and φ, ψ are closure operators on P, we say φ ψ if and only if φ(x) ≤ ψ(x) for all x ∈ P. If P is a lattice, then the set of closure operators on P is again a lattice, via (φ ∧ ψ)(x) = φ(x) ∧ ψ(x) (φ ∨ ψ)(x) =

  • {ν(x) | φ ≤ ν and ψ ≤ ν}

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Question

Let G be a group: sub(G) is the lattice of all subgroups of G.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Question

Let G be a group: sub(G) is the lattice of all subgroups of G. Let L be a lattice: c.o.(L) is the lattice of all closure operators

  • n L.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Question

Let G be a group: sub(G) is the lattice of all subgroups of G. Let L be a lattice: c.o.(L) is the lattice of all closure operators

  • n L.

Question Let G be a group. When is c.o(sub(G)) isomorphic to sub(K) for some group K?

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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But wait....

The lattice of subgroups of a group is always algebraic. This reflects the fact that a group is completely determined by its finitely generated subgroups.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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But wait....

The lattice of subgroups of a group is always algebraic. This reflects the fact that a group is completely determined by its finitely generated subgroups. If L is an infinite lattice, then c.o.(L) is not always an algebraic lattice, and so would be disqualified a priori.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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But wait....

The lattice of subgroups of a group is always algebraic. This reflects the fact that a group is completely determined by its finitely generated subgroups. If L is an infinite lattice, then c.o.(L) is not always an algebraic lattice, and so would be disqualified a priori. Definition Let G be a group. A closure operator φ on sub(G) is algebraic if for every H ≤ G, we have φ(H) =

  • K≤H

K f.g.

φ(K).

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Stick to algebraic closure operators

Theorem (Kilpack) The lattice of all algebraic closure operators on a lattice L is an algebraic lattice; it is a lower subsemilattice of the lattice of all closure operators on L.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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The new question

Question Let G be a group. When is the lattice of all algebraic closure

  • perators on sub(G), aco(sub(G)), isomorphic to sub(K) for

some group K?

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Easy case for yes

Theorem Let G be a cyclic group Cpn for some prime p. Then aco(sub(G)) is isomorphic to sub(Cq1···qn), where q1 < · · · < qn are distinct primes.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Easy case for yes

Theorem Let G be a cyclic group Cpn for some prime p. Then aco(sub(G)) is isomorphic to sub(Cq1···qn), where q1 < · · · < qn are distinct primes.

  • Proof. A closure operator on sub(G) is completely determined

by specifying which proper subgroups are closed.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Easy case for yes

Theorem Let G be a cyclic group Cpn for some prime p. Then aco(sub(G)) is isomorphic to sub(Cq1···qn), where q1 < · · · < qn are distinct primes.

  • Proof. A closure operator on sub(G) is completely determined

by specifying which proper subgroups are closed. This corresponds to the subgroup of subsets of {1, p, p2, . . . , pn−1}.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Easy case for yes

Theorem Let G be a cyclic group Cpn for some prime p. Then aco(sub(G)) is isomorphic to sub(Cq1···qn), where q1 < · · · < qn are distinct primes.

  • Proof. A closure operator on sub(G) is completely determined

by specifying which proper subgroups are closed. This corresponds to the subgroup of subsets of {1, p, p2, . . . , pn−1}. The subgroups of Cq1···qn correspond to divisors of q1 · · · qn, which correspond to subsets of {q1, . . . , qn}.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Easy case for yes

Theorem Let G be a cyclic group Cpn for some prime p. Then aco(sub(G)) is isomorphic to sub(Cq1···qn), where q1 < · · · < qn are distinct primes.

  • Proof. A closure operator on sub(G) is completely determined

by specifying which proper subgroups are closed. This corresponds to the subgroup of subsets of {1, p, p2, . . . , pn−1}. The subgroups of Cq1···qn correspond to divisors of q1 · · · qn, which correspond to subsets of {q1, . . . , qn}. Hence aco(sub(Cpn)) ∼ = sub(Cq1···qn).

  • Martha Kilpack and Arturo Magidin

Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • Martha Kilpack and Arturo Magidin

Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φ0,x,y,1

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φ0,y,1

φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φ0,x,1

φ0,y,1 φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ■■■■■■■■■

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φ0,1

φ0,x,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

φ0,y,1

■■■■■■■■■

φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ■■■■■■■■■

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φ0,1

φ0,x,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

φ0,y,1

■■■■■■■■■

φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ■■■■■■■■■

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 34

Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φx,1

φ0,1 φ0,x,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

φ0,y,1

■■■■■■■■■

φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ■■■■■■■■■

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 35

Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φx,1

φ0,1 φy,1 φ0,x,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

φ0,y,1

■■■■■■■■■

φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ■■■■■■■■■

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Closure operators on sub(Cpq)

Lattice Closure operators 1 x

  • y

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

  • φ1

φx,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

φ0,1 φy,1

■■■■■■■■■■

φ0,x,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

φ0,y,1

■■■■■■■■■

φ0,x,y,1

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ■■■■■■■■■

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Two easy exercises

Proposition If G has a unique maximal subgroup, then G is cyclic of prime power order.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Two easy exercises

Proposition If G has a unique maximal subgroup, then G is cyclic of prime power order. In particular, its lattice of subgroups is a finite chain.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Two easy exercises

Proposition If G has a unique maximal subgroup, then G is cyclic of prime power order. In particular, its lattice of subgroups is a finite chain. Proposition If G has exactly two maximal subgroups, then G is cyclic of

  • rder paqb for some distinct primes p and q, a, b > 0.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

t t t t t t t t t

φ1,q,pq

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

❏❏❏❏❏❏❏❏❏ t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 41

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

t t t t t t t t t

φ1,q,pq

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

❏❏❏❏❏❏❏❏❏ t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 42

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

ℓ

t t t t t t t t t

φ1,q,pq

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

ℓ

❏❏❏❏❏❏❏❏❏ t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 43

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

ℓ

t t t t t t t t t

φ1,q,pq

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

ℓ

❏❏❏❏❏❏❏❏❏ t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 44

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

ℓ

t t t t t t t t t

φ1,q,pq

k

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

ℓ

❏❏❏❏❏❏❏❏❏

k

t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 45

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

ℓ

t t t t t t t t t

φ1,q,pq

k

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

ℓ

❏❏❏❏❏❏❏❏❏

k

t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 46

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

t t t t t t t t t t

φ1,pq φq,pq

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

ℓ k

t t t t t t t t t

φ1,q,pq

k ℓ

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

ℓ

❏❏❏❏❏❏❏❏❏

k

t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

slide-47
SLIDE 47

Is this a subgroup lattice?

Subgroups Closure operators Cpq Cp

⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

Cq

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

{e}

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤

φpq φp,pq

prime

t t t t t t t t t t

φ1,pq

prime

φq,pq

prime

❏❏❏❏❏❏❏❏❏❏

φ1,p,pq

ℓ k

t t t t t t t t t

φ1,q,pq

k ℓ

❏❏❏❏❏❏❏❏❏

φ1,p,q,pq

ℓ

❏❏❏❏❏❏❏❏❏

k

t t t t t t t t t

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 48

Not a subgroup lattice

Theorem Let p and q be distinct primes. Then the lattice of (algebraic) closure operators on sub(Cpq) is not isomorphic to a subgroup lattice.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 49

The key constructions

Idea: Find closure operators φ and ψ with ψ < φ, and such that for all η, η < φ = ⇒ η ≤ ψ.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 50

The key constructions

Idea: Find closure operators φ and ψ with ψ < φ, and such that for all η, η < φ = ⇒ η ≤ ψ. If the closure operator lattice is a subgroup lattice, then ↓ (φ) must be a finite chain.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 51

The key constructions

Idea: Find closure operators φ and ψ with ψ < φ, and such that for all η, η < φ = ⇒ η ≤ ψ. If the closure operator lattice is a subgroup lattice, then ↓ (φ) must be a finite chain. Given a subgroup H of prime order, let φ(K) = K, H.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

slide-52
SLIDE 52

The key constructions

Idea: Find closure operators φ and ψ with ψ < φ, and such that for all η, η < φ = ⇒ η ≤ ψ. If the closure operator lattice is a subgroup lattice, then ↓ (φ) must be a finite chain. Given a subgroup H of prime order, let φ(K) = K, H. Then let ψ map {e} to itself, and K to K, H otherwise.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 53

The key constructions

Idea: Find closure operators φ and ψ with ψ < φ, and such that for all η, η < φ = ⇒ η ≤ ψ. If the closure operator lattice is a subgroup lattice, then ↓ (φ) must be a finite chain. Given a subgroup H of prime order, let φ(K) = K, H. Then let ψ map {e} to itself, and K to K, H otherwise. These two satisfy the condition above.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 54

Main exclusion theorem

Theorem (Main exclusion theorem) Suppose that G is a group, H is a subgroup of prime order, and there exist subgroups M and N such that: H M and H N; and M and N are incomparable; then aco(sub(G)) is not a subgroup lattice.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 55

Main exclusion theorem

Theorem (Main exclusion theorem) Suppose that G is a group, H is a subgroup of prime order, and there exist subgroups M and N such that: H M and H N; and M and N are incomparable; then aco(sub(G)) is not a subgroup lattice. Corollary If G has at least three distinct subgroups of prime order, then aco(sub(G)) is not a subgroup lattice.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 56

Quotients

Theorem Let N ⊳ G be a finitely generated normal subgroup. Define φN(K) =

  • G

if N ≤ K; K

  • therwise.

Then ↓ (φN) in aco(sub(G)) ∼ = aco(↑ (N) in sub(G)). In particular, if aco(sub(G)) is a subgroup lattice, then aco(sub(G/N)) is a subgroup lattice.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 57

Finite case

Theorem Let G be a finite group. Then aco(sub(G)) is a subgroup lattice if and only if G is a cyclic group of prime power order.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 58

Finite case

Theorem Let G be a finite group. Then aco(sub(G)) is a subgroup lattice if and only if G is a cyclic group of prime power order.

  • Sketch. Must have order pn or paqb, and either exactly one or

exactly two subgroups of prime order. Use induction on n and

  • n a + b.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 59

Infinite mixed case

Theorem Let G be a group that has nontrivial elements of finite and of infinite order. Then aco(sub(G)) is not a subgroup lattice.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 60

Infinite mixed case

Theorem Let G be a group that has nontrivial elements of finite and of infinite order. Then aco(sub(G)) is not a subgroup lattice.

  • Proof. Let H be a subgroup of prime order, and let x be an

element of infinite order.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 61

Infinite mixed case

Theorem Let G be a group that has nontrivial elements of finite and of infinite order. Then aco(sub(G)) is not a subgroup lattice.

  • Proof. Let H be a subgroup of prime order, and let x be an

element of infinite order. Then M = x2 and N = x3 are incomparable, and do not contain H. By the Main Exclusion Theorem, aco(sub(G)) is not a subgroup lattice.

  • Martha Kilpack and Arturo Magidin

Closure operators on subgroup lattices

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SLIDE 62

Infinite torsion group

If G is infinite torsion, there are at most two primes p and q such that every element is of order paqb.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 63

Infinite torsion group

If G is infinite torsion, there are at most two primes p and q such that every element is of order paqb. If G is a p-group, one can show it has a unique subgroup of

  • rder pn for all n, so G ∼

= Zp∞.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 64

Infinite torsion group

If G is infinite torsion, there are at most two primes p and q such that every element is of order paqb. If G is a p-group, one can show it has a unique subgroup of

  • rder pn for all n, so G ∼

= Zp∞. Otherwise, we can reduce to the case where G has a subgroup isomorphic to Zp∞ and a subgroup H of order q = p.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 65

An infinite chain

Cp2, H ... Cp2

① ① ① ① ① ① ① ① ① ① ①

Cp, H ...

❑❑❑❑❑❑❑❑❑❑

Cp

❋❋❋❋❋❋❋❋❋❋❋❋ s s s s s s s s s s s s s s

H ...

  • {e}

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 66

An infinite chain

Cp2, H ... φ Cp2

  • ①

① ① ① ① ① ① ① ① ① ①

Cp, H ...

❑❑❑❑❑❑❑❑❑❑

Cp

❋❋❋❋❋❋❋❋❋❋❋❋

  • s

s s s s s s s s s s s s s

H ...

  • {e}

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼

  • ✉

✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 67

An infinite chain

Cp2, H ... φ Cp2

  • ①

① ① ① ① ① ① ① ① ① ①

Cp, H ...

❑❑❑❑❑❑❑❑❑❑

ψ0 Cp

❋❋❋❋❋❋❋❋❋❋❋❋

  • s

s s s s s s s s s s s s s

H ...

  • {e}

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 68

An infinite chain

Cp2, H ... φ Cp2

  • ①

① ① ① ① ① ① ① ① ① ①

Cp, H ...

❑❑❑❑❑❑❑❑❑❑

ψ0 Cp

❋❋❋❋❋❋❋❋❋❋❋❋ s s s s s s s s s s s s s s

H ...

  • ψ1

{e}

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 69

An infinite chain

Cp2, H ... φ Cp2

① ① ① ① ① ① ① ① ① ① ①

Cp, H ...

❑❑❑❑❑❑❑❑❑❑

ψ0 Cp

❋❋❋❋❋❋❋❋❋❋❋❋ s s s s s s s s s s s s s s

H ...

  • ψ1

{e}

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

ψ2

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 70

An infinite chain

Cp2, H ... φ Cp2

① ① ① ① ① ① ① ① ① ① ①

Cp, H ...

❑❑❑❑❑❑❑❑❑❑

ψ0 Cp

❋❋❋❋❋❋❋❋❋❋❋❋ s s s s s s s s s s s s s s

H ...

  • ψ1

{e}

▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

ψ2

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 71

Groups with torsion

Theorem Let G be a group (finite or infinite) that has a nontrivial element

  • f finite order. Then aco(sub(G)) is a subgroup lattice if and only

if G is cyclic of prime power order or isomorphic to the Prüfer p-group Zp∞.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 72

Open cases

What about torsionfree groups?

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 73

Open cases

What about torsionfree groups? Most of the constructions require a nontrivial minimal subgroup.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 74

Open cases

What about torsionfree groups? Most of the constructions require a nontrivial minimal subgroup. If G has a nontrivial finitely generated normal abelian subgroup, then aco(sub(G)) is not a subgroup lattice.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 75

Open cases

What about torsionfree groups? Most of the constructions require a nontrivial minimal subgroup. If G has a nontrivial finitely generated normal abelian subgroup, then aco(sub(G)) is not a subgroup lattice. If G has a finitely generated normal subgroup N, then G/N must be torsionfree, cyclic of prime power order, or isomorphic to Zp∞.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 76

Conjecture

Conjecture Let G be a group. Then aco(sub(G)) is a subgroup lattice if and

  • nly if G is cyclic of prime power order of isomorphic to the

Prüfer p-group.

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices

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SLIDE 77

Thank you for your attention!

Martha Kilpack and Arturo Magidin Closure operators on subgroup lattices