Torsion Units in Integral Group Rings Andreas B achle Vrije - PowerPoint PPT Presentation

Torsion Units in Integral Group Rings Andreas B¨ achle Vrije Universiteit Brussel Computational Methods for Representations and Group Rings University of Stuttgart, February 18 and 19, 2016

Notations G finite group

Notations G finite group Z G integral group ring of G

Notations G finite group Z G integral group ring of G � � � = � ε augmentation map of Z G , i.e. ε r g g r g . g ∈ G g ∈ G

Notations G finite group Z G integral group ring of G � � � = � ε augmentation map of Z G , i.e. ε r g g r g . g ∈ G g ∈ G U( Z G ) group of units of Z G

Notations G finite group Z G integral group ring of G � � � = � ε augmentation map of Z G , i.e. ε r g g r g . g ∈ G g ∈ G U( Z G ) group of units of Z G V( Z G ) group of units of Z G of augmentation 1 aka normalized units.

Notations G finite group Z G integral group ring of G � � � = � ε augmentation map of Z G , i.e. ε r g g r g . g ∈ G g ∈ G U( Z G ) group of units of Z G V( Z G ) group of units of Z G of augmentation 1 aka normalized units. U( Z G ) = ± V( Z G ).

Questions How strong is the connection between the group G and the set of torsion subgroups in V( Z G )?

Questions How strong is the connection between the group G and the set of torsion subgroups in V( Z G )? For example, for finite X ≤ V( Z G ) we have � (ˇ ◮ | X | � � | G | Zmud, Kurenno˘ ı 1967)

Questions How strong is the connection between the group G and the set of torsion subgroups in V( Z G )? For example, for finite X ≤ V( Z G ) we have � (ˇ ◮ | X | � � | G | Zmud, Kurenno˘ ı 1967) � � ◮ exp X � exp G (Cohn, Livingstone, 1965)

Questions How strong is the connection between the group G and the set of torsion subgroups in V( Z G )? For example, for finite X ≤ V( Z G ) we have � (ˇ ◮ | X | � � | G | Zmud, Kurenno˘ ı 1967) � � ◮ exp X � exp G (Cohn, Livingstone, 1965) ◮ Let X ≤ V( Z G ) finite, ◮ is X ∼ RG H ≤ G ?

Questions How strong is the connection between the group G and the set of torsion subgroups in V( Z G )? For example, for finite X ≤ V( Z G ) we have � (ˇ ◮ | X | � � | G | Zmud, Kurenno˘ ı 1967) � � ◮ exp X � exp G (Cohn, Livingstone, 1965) ◮ Let X ≤ V( Z G ) finite, ◮ is X ∼ RG H ≤ G ? ◮ is X ≃ H ≤ G ?

Questions How strong is the connection between the group G and the set of torsion subgroups in V( Z G )? For example, for finite X ≤ V( Z G ) we have � (ˇ ◮ | X | � � | G | Zmud, Kurenno˘ ı 1967) � � ◮ exp X � exp G (Cohn, Livingstone, 1965) ◮ Let X ≤ V( Z G ) finite, ◮ is X ∼ RG H ≤ G ? ◮ is X ≃ H ≤ G ? ◮ Z G ≃ Z H = ⇒ ? G ≃ H ?

Example: S 3 3 � Z � 3 Z ∼ ϕ : Z S 3 → Z ⊕ Z ⊕ Z Z 2 3 � − 2 − 3 � 1 � � . (1 , − 1 , ) ∈ ϕ ( S 3 ) (1 , − 1 , ) . − 1 1 2

Zassenhaus Conjcetures Source: S. Sehgal, Torsion units in group rings. Methods in ring theory (Antwerp, 1983)

(First) Zassenhaus Conjecture (Zassenhaus 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g .

(First) Zassenhaus Conjecture (Zassenhaus 1960s) (ZC1) For u ∈ V( Z G ) of finite order there exist x ∈ U( Q G ) and g ∈ G such that x − 1 ux = g . � abelian groups (Higman 1939) � A 5 (Luthar, Passi 1989) � S 5 (Luthar, Trama 1991) � nilpotent groups (Weiss 1991) � groups of order at most 71 (H¨ ofert 2004) � PSL(2 , 7), PSL(2 , 11), PSL(2 , 13) (Hertweck 2004) � A 6 ≃ PSL(2 , 9) (Hertweck 2007) � metacyclic groups (Hertweck 2008) � PSL(2 , 8), PSL(2 , 17) (Gildea; Kimmerle, Konovalov 2012) � cyclic-by-abelian (Caicedo, Margolis, del R´ ıo 2013)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle 2006) (PQ) Γ( G ) = Γ(V( Z G ))?

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � solvable groups (H¨ ofert, Kimmerle 2006)

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � solvable groups (H¨ ofert, Kimmerle 2006) � half of the sporadic simple groups (Bovdi, Konovalov, et. al. 2005 – )

The prime graph (or Gruenberg-Kegel graph ) of a group H is the undirected loop-free graph Γ( H ) with ◮ Vertices: primes p , s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H Prime graph question (Kimmerle 2006) (PQ) Γ( G ) = Γ(V( Z G ))? Clearly: (ZC1) = ⇒ (PQ) � solvable groups (H¨ ofert, Kimmerle 2006) � half of the sporadic simple groups (Bovdi, Konovalov, et. al. 2005 – ) � PSL(2 , p ) (Hertweck 2006)

Let x ∈ G and x G be its conjugacy class. Let u = � u g g ∈ Z G . g ∈ G Then � ε x ( u ) = u g g ∈ x G is the partial augmentation of u at (the conjugacy class of) x .

Let x ∈ G and x G be its conjugacy class. Let u = � u g g ∈ Z G . g ∈ G Then � ε x ( u ) = u g g ∈ x G is the partial augmentation of u at (the conjugacy class of) x . Proposition (Marciniak, Ritter, Sehgal, Weiss 1987) Let u ∈ V( Z G ) be of finite order n. Then u is conjugate to an element of G in Q G ⇔ ε g ( u d ) ≥ 0 for all g ∈ G and all d | n.

Let x ∈ G and x G be its conjugacy class. Let u = � u g g ∈ Z G . g ∈ G Then � ε x ( u ) = u g g ∈ x G is the partial augmentation of u at (the conjugacy class of) x . Proposition (Marciniak, Ritter, Sehgal, Weiss 1987) Let u ∈ V( Z G ) be of finite order n. Then u is conjugate to an element of G in Q G ⇔ ε g ( u d ) ≥ 0 for all g ∈ G and all d | n. For u ∈ Z G a normalized torsion unit of order n � = 1 we have ◮ ε 1 ( u ) = 0 (Berman 1955; Higman 1939) ◮ ε x ( u ) = 0 if o ( x ) ∤ n (Hertweck 2004)

Theorem (Luthar, Passi 1989; Hertweck 2004) ◮ u ∈ Z G torsion unit of order n ◮ F splitting field for G with p = char( F ) ∤ n ◮ χ a (Brauer) character of F-representation D of G ◮ ζ ∈ C primitive n-th root of unity ◮ ξ ∈ F corresponding n-th root of unity

Theorem (Luthar, Passi 1989; Hertweck 2004) ◮ u ∈ Z G torsion unit of order n ◮ F splitting field for G with p = char( F ) ∤ n ◮ χ a (Brauer) character of F-representation D of G ◮ ζ ∈ C primitive n-th root of unity ◮ ξ ∈ F corresponding n-th root of unity Multiplicity µ ℓ ( u , χ, p ) of ξ ℓ as an eigenvalue of D ( u ) is given by 1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) n d | n

µ ℓ ( u , χ, p ) = 1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k d | k

µ ℓ ( u , χ, p ) = 1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k d | k = 1 Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) +1 � k Tr Q ( ζ ) / Q ( χ ( u ) ζ − ℓ ) k d | k d � =1

µ ℓ ( u , χ, p ) = 1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k d | k = 1 +1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k Tr Q ( ζ ) / Q ( χ ( u ) ζ − ℓ ) k d | k d � =1 � �� � =: a χ,ℓ

µ ℓ ( u , χ, p ) = 1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k d | k = 1 +1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k Tr Q ( ζ ) / Q ( χ ( u ) ζ − ℓ ) k d | k d � =1 � �� � =: a χ,ℓ As χ ( u ) = � ε C ( u ) χ ( C ), C

µ ℓ ( u , χ, p ) = 1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k d | k = 1 +1 � Tr Q ( ζ d ) / Q ( χ ( u d ) ζ − d ℓ ) k Tr Q ( ζ ) / Q ( χ ( u ) ζ − ℓ ) k d | k d � =1 � �� � =: a χ,ℓ As χ ( u ) = � ε C ( u ) χ ( C ), we obtain linear equations C µ ℓ ( u , χ, p ) = t C 1 ε C 1 ( u ) + t C 2 ε C 2 ( u ) + ... + t C h ε C h ( u ) + a χ,ℓ . for the partial augmentations ε C i ( u ), with “known” coefficients t C j , a χ,ℓ and µ ℓ ( u , χ, p ).

Commercial break: The HeLP-package for GAP . Using 4ti2 and soon also Normaliz .

Theorem (B., Caicedo 2015) If G is an almost simple group with socle A n , 11 ≤ n ≤ 17 , then (PQ) has an affirmative answer for G.