Units of integral group rings

AbbreviationThe group \(C_2 \times C_2 \times C_2 \times C_6\)#✓fully proved

The abelian group \(G = C_2 \times C_2 \times C_2 \times C_6\), the product of cyclic groups of orders \(2\), \(2\), \(2\) and \(6\), written additively as \(\mathbb{Z}/2 \times \mathbb{Z}/2 \times \mathbb{Z}/2 \times \mathbb{Z}/6\).

abbrev OWRGroup : Type := ZMod 2 × ZMod 2 × ZMod 2 × ZMod 6

Sources

• OWR-11129-005 (Computational Group Theory (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/37).

Used by 2

Used byExotic unit conjecture for \(C_2 \times C_2 \times C_2 \times C_6\) · The integral group ring of \(C_2 \times C_2 \times C_2 \times C_6\)

AbbreviationThe integral group ring of \(C_2 \times C_2 \times C_2 \times C_6\)#✓fully proved

The integral group ring \(\mathbb{Z}G\) of \(G = C_2 \times C_2 \times C_2 \times C_6\): finitely supported formal \(\mathbb{Z}\)-linear combinations of elements of \(G\), with the product extending the group operation of \(G\) bilinearly.

abbrev OWRGroupRing : Type := AddMonoidAlgebra ℤ OWRGroup

Sources

• OWR-11129-005 (Computational Group Theory (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/37).

Uses 1 · Used by 1

Statement usesThe group \(C_2 \times C_2 \times C_2 \times C_6\)

Used byExotic unit conjecture for \(C_2 \times C_2 \times C_2 \times C_6\)

DefinitionExotic unit conjecture for \(C_2 \times C_2 \times C_2 \times C_6\)#✓fully proved

Let \(G = C_2 \times C_2 \times C_2 \times C_6\). The statement that the integral group ring \(\mathbb{Z}G\) contains an exotic unit: a unit \(u\) with \(u \ne g\) and \(u \ne -g\) for every \(g \in G\).

def conjecture : Prop :=
  ∃ u : OWRGroupRingˣ, ∀ g : OWRGroup, ∀ s : ℤˣ,
    (↑u : OWRGroupRing) ≠ AddMonoidAlgebra.single g (↑s : ℤ)

Sources

• the open question of OWR-11129-005 (Computational Group Theory (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/37).

Uses 2 · Used by 1

Statement usesThe group \(C_2 \times C_2 \times C_2 \times C_6\) · The integral group ring of \(C_2 \times C_2 \times C_2 \times C_6\)

Used byNo exotic units in the integral group ring of \(C_2 \times C_2 \times C_2 \times C_6\)

TheoremNo exotic units in the integral group ring of \(C_2 \times C_2 \times C_2 \times C_6\)#●ready to prove

Let \(G = C_2 \times C_2 \times C_2 \times C_6\). Every unit \(u\) of the integral group ring \(\mathbb{Z}G\) is trivial: \(u = g\) or \(u = -g\) for some \(g \in G\).

Classical: G. Higman (1940) showed the integral group ring of an abelian group of exponent dividing 4 or 6 has only the units \(\pm g\). Via the cut-group criterion in Bächle (arXiv:1701.04347, Prop. 2.2), \(C_2^3 \times C_6\) (exponent 6) is an abelian cut group, so every unit of \(\mathbb{Z}G\) is trivial and no exotic unit exists.

public theorem_wanted conjecture_refuted : ¬ conjecture

Sources

• Andreas Bächle, Integral group rings of solvable groups with trivial central units, Forum Math. 30 (2018), arXiv:1701.04347, https://arxiv.org/abs/1701.04347.

Uses 1

Statement usesExotic unit conjecture for \(C_2 \times C_2 \times C_2 \times C_6\)

3 fully proved1 ready to prove

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