The Multiple Holomorph of a Finitely-Generated Abelian Group

. The multiple holomorph of a ﬁnitely-generated abelian group .

Andrea Caranti1 & Francesca Dalla Volta2

1Dipartimento di Matematica Università degli Studi di Trento

2Dipartimento di Matematica e Applicazioni Università degli Studi di Milano–Bicocca

Lincoln, 31 August 2016

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 1 / 28 . Groups, Rings, and Their Automorphisms .

Andrea Caranti1 & Francesca Dalla Volta2

1Dipartimento di Matematica Università degli Studi di Trento

2Dipartimento di Matematica e Applicazioni Università degli Studi di Milano–Bicocca

Lincoln, 31 August 2016

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 2 / 28 Plan

1. Holomorphs The holomorph Same holomorph The multiple holomorph

2. Groups and rings Regular abelian groups and rings Finitely generated abelian groups Arguments Examples

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 3 / 28 Holomorphs The holomorph Holomorphs

The holomorph of a group G is the natural semidirect product

Aut(G)G

of G by its automorphism group Aut(G). Consider the right regular representation ρ : G → S(G) g 7→ (x 7→ x + g) where S(G) is the group of permutations of the set G. (Our groups (G, +) will be abelian) Then it is well-known that

NS(G)(ρ(G)) = Aut(G)ρ(G) is (isomorphic to) the holomorph Hol(G) of G. More generally, if N is any regular subgroup of S(G), its holomorph will be (isomorphic to) NS(G)(N). Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 5 / 28 Holomorphs Same holomorph Same holomorph

So you may want to say that G and N have the same holomorph if

Hol(N) = NS(G)(N) = NS(G)(ρ(G)) = Hol(G).

Note that the dihedral and quaternionic groups of the same order have the same holomorph. We will be considering a stronger requirement and a weaker one.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 7 / 28 Holomorphs The multiple holomorph The multiple holomorph

G. A. Miller On the multiple holomorphs of a group Math. Ann. 66 (1908), no. 1, 133–142 Miller considered the multiple holomorph of G

NS(G)(Hol(G)) = NS(G)(NS(G)(ρ(G))).

This has the property that the quotient

T (G) = NS(G)(Hol(G))/ Hol(G)

acts regularly by conjugation on the set ∼ H(G) = {N ≤ S(G): N is regular, Hol(N) = Hol(G), and N = G}.

This is the stronger requirement.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 9 / 28 Holomorphs The multiple holomorph Dihedral and quaternionic

Timothy Kohl Multiple holomorphs of dihedral and quaternionic groups Comm. Algebra 43 (2015), no. 10, 4290–4304 Kohl has worked out the set ∼ H(G) = {N ≤ S(G): N is regular, Hol(N) = Hol(G), and N = G}.

and the group T (G) = NS(G)(Hol(G))/ Hol(G) for G a dihedral and quaternionic group. Long story short: T (G) is elementary abelian.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 10 / 28 Holomorphs The multiple holomorph Finitely generated abelian groups

W. H. Mills Multiple holomorphs of ﬁnitely generated abelian groups Trans. Amer. Math. Soc. 71 (1951), 379–392 Mills determined ∼ H(G) = {N ≤ S(G): N is regular, Hol(N) = Hol(G), and N = G}.

and the group T (G) = NS(G)(Hol(G))/ Hol(G) for ﬁnitely generated abelian groups G. In a nutshell, T (G) is elementary abelian, of order 1, 2 or 4. We are redoing Mills’s work via a diﬀerent approach.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 11 / 28 Groups and rings Regular abelian groups and rings Regular abelian groups and rings

A. Caranti, F. Dalla Volta, and M. Sala Abelian regular subgroups of the aﬃne group and radical rings Publ. Math. Debrecen 69 (2006), no. 3, 297–308 When G is abelian, the regular abelian subgroups N ≤ Hol(G) can be described in terms of commutative ring structures (G, +, ·) on (G, +), so that (G, +, ·) is a radical ring. N ⊴ Hol(G) (i.e. Hol(N) ≥ Hol(G), our weaker requirement) translates into the fact that all automorphisms of the group (G, +) are also automorphisms of the ring (G, +, ·). The elements of T (G) = N (Hol(G))/ Hol(G) are (classes of) S(G) ∼ isomorphisms between (G, +) and the circle group (G, ◦) = N, where

x ◦ y = x + y + xy.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 13 / 28 Groups and rings Finitely generated abelian groups Finitely generated abelian groups

ι : x 7→ −x is an automorphism of the group (G, +). Thus it is also an automorphism of the ring (G, +, ·). Therefore ι(xy) = ι(x)ι(y), that is −xy = (−x)(−y) = xy, or 2xy = 0. In particular, if x has odd order d, then it lies in the annihilator

xy = d(xy) = (dx)y = 0.

In these rings we also have xyz = 0.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 15 / 28 Groups and rings Finitely generated abelian groups Doing it with rings

So our goal becomes: Given a ﬁnitely generated abelian group (G, +), ﬁnd the (commutative) ring structures (G, +, ·) with 2xy = 0, xyz = 0 such that all automorphisms of (G, +) are also automorphisms of (G, +, ·). This will yield the regular abelian subgroups N of S(G) such that

Hol(N) = NS(G)(N) ≥ NS(G)(ρ(G)) = Hol(G).

∼ ∼ Then check for which of these rings we have that N = (G, ◦) = G, as this will yield the elements of ∼ H(G) = { N ≤ S(G): N is regular, N = G and Hol(N) = Hol(G) } . Here x ◦ y = x + y + xy.

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 16 / 28 Groups and rings Finitely generated abelian groups Notation

So let G = F × H, where F is free abelian, H is a ﬁnite abelian 2-group. Let ∏ F = n ⟨ z ⟩, ∏i=1 i m ⟨ ⟩ | | ≥ | | H = i=1 xi , with xi xi+1 .

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 17 / 28 Groups and rings Arguments Sample arguments

G = F × H, with ∏ F free abelian, F = n ⟨ z ⟩, i=1 i ∏ m ⟨ ⟩ | | ≥ | | H a finite abelian 2-group, H = i=1 xi , with xi xi+1 . . Lemma . 2 .If F ≠ 0, then H = { gh : g, h ∈ H } = 0. . Proof. . All products are involutions: z1xj ∈ H is fixed by the automorphism of G that leaves each zi , xi fixed, except z1 7→ z1 + xi . Therefore z1xj = (z1 + xi )xj = z1xj + xi xj ,

.whence xi xj = 0.

.Go to examples Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 19 / 28 Groups and rings Arguments Sample arguments

G = F × H, with ∏ F free abelian, F = n ⟨ z ⟩, i=1 i ∏ H a finite abelian 2-group, H = m ⟨ x ⟩, with |x | ≥ |x |. . i=1 i i i+1 Lemma . .If n ≥ 3, then F 2 = 0. . Proof. . Let i, j, k be distinct indices for the z. All products are involutions: zi zk ∈ H is fixed by the automorphism of G that leaves each zi , xi fixed, except zk 7→ zk + zj . Therefore zi zk = zi (zk + zj ) = zi zk + zi zj ,

.whence zi zj = 0.

.Go to examples Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 20 / 28 Groups and rings Arguments Sample arguments

G = F × H, with ∏ n ⟨ ⟩ F free abelian, F = i=1 zi , ∏ m ⟨ ⟩ | | ≥ | | . H a ﬁnite abelian 2-group, H = i=1 xi , with xi xi+1 . Lemma . 2 2 .If F ≠ 0, then F is the unique characteristic minimal subgroup of H. . Proof. . Extend the automorphisms of H to G so that they act trivially on F. All products are involutions: F 2 ≤ H is ﬁxed elementwise by all automorphisms of H. Thus each (minimal) subgroup of F 2 is characteristic in H. But the 2-group H can have at most one characteristic minimal subgroup, and this occurs when H is cyclic, or |x1| > |x2|. .

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 21 / 28 Groups and rings Examples Non isomorphic

G = F × H, with ∏ F free abelian, F = n ⟨ z ⟩, i=1 i ∏ m ⟨ ⟩ | | ≥ | | H a ﬁnite abelian 2-group, H = i=1 xi , with xi xi+1 . 2 G = H = ⟨ x1 ⟩, of order 4. Ring structure given by x = 2x1. Then in ∼ 1 (G, ◦) = N, with x ◦ y = x + y + xy. we have ◦ 2 x1 x1 = x1 + x1 + x1 = 2x1 + 2x1 = 0.

Thus (G, ◦) is elementary abelian. Therefore N ⊴ Hol(G) = NS(G)(ρ(G)), but Hol(N) = NS(G)(N) > NS(G)(ρ(G)) = Hol(G), because the ﬁrst group has order 4 · 6, while the second one has order 4 · 2.

.Next

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 23 / 28 Groups and rings Examples Four 2-groups

G = F × H, with ∏ F free abelian, F = n ⟨ z ⟩, i=1 i ∏ m ⟨ ⟩ | | ≥ | | H a ﬁnite abelian 2-group, H = i=1 xi , with xi xi+1 . F = 0, so G = H is a ﬁnite 2-group. ∼ G = ⟨ x1 ⟩ × ⟨ x2 ⟩, with |x1| > 4 (to ensure G = N) and |x1| > |x2|. Write ti for the unique involution of ⟨ xi ⟩. Four possibilities for the ring . 2 2 1 x1 = 0, x2 = 0, x1x2 = 0. . 2 2 2 x1 = 0, x2 = 0, x1x2 = t1. . 2 2 3 x1 = t1, x2 = 0, x1x2 = 0. . 2 2 4 x1 = t1, x2 = 0, x1x2 = t1.

.Next

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 24 / 28 Groups and rings Examples Four mixed groups

G = F × H, with ∏ n ⟨ ⟩ F free abelian, F = i=1 zi , ∏ m ⟨ ⟩ | | ≥ | | H a ﬁnite abelian 2-group, H = i=1 xi , with xi xi+1 .

n = m = 1, so as a group G = ⟨ z1 ⟩ × ⟨ x1 ⟩, and |x1| ≥ 4. Let t1 be the unique involution in ⟨ x1 ⟩. Four possibilities for the ring . 2 2 1 z1 = 0, z1x1 = 0, x1 = 0. . 2 2 2 z1 = t1, z1x1 = 0, x1 = 0. . 2 2 3 z1 = t1, z1x1 = t1, x1 = 0. . 2 2 4 z1 = 0, z1x1 = t1, x1 = 0. |x1| ≥ 4, as if x1 = t1 has order 2, then

0 = z1(z1x1) = z1t1 = z1x1 = t1.

So if |x1| = 2 there are only two cases, not four. .Next Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 25 / 28 Groups and rings Examples A larger H2

G = F × H, with ∏ F free abelian, F = n ⟨ z ⟩, i=1 i ∏ m ⟨ ⟩ | | ≥ | | H a ﬁnite abelian 2-group, H = i=1 xi , with xi xi+1 . F = 0, so G = H is a ﬁnite 2-group. G = ⟨ x1 ⟩ × ⟨ x2 ⟩, with |x1| = |x2| > 4. Write ti for the unique involution of ⟨ xi ⟩. Two possibilities for the ring 1. G is the zero ring. . 2 2 2 x1 = t1, x2 = t2, x1x2 = t1 + t2.

.Next

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 26 / 28 Groups and rings Examples A larger FH

G = F × H, with ∏ F free abelian, F = n ⟨ z ⟩, i=1 i ∏ m ⟨ ⟩ | | ≥ | | H a ﬁnite abelian 2-group, H = i=1 xi , with xi xi+1 . n = 1, m > 1, so as a group

G = ⟨ z1 ⟩ × ⟨ x1 ⟩ × · · · × ⟨ xm ⟩,

and we choose |x1| = ··· = |xm| ≥ 4. Let ti be the unique involution in ⟨ xi ⟩. Two possible rings . 2 1 z1 = 0, z1x1 = 0, xi xj = 0. . 2 2 z1 = 0, z1xi = ti , xi xj = 0.

.Skip to end

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 27 / 28 Groups and rings Examples Thanks

That’s All, Thanks!

Caranti & Dalla Volta (unitn & unimib) Multiple Holomorph Lincoln, 31 August 2016 28 / 28