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RATIONALITY OF GROUPS AND CENTERS OF INTEGRAL GROUP RINGS Andreas Bchle Groups St Andrews 2017 1 NOTATION. G finite gr oup Z G integral group ring of G U ( Z G ) group of units of Z G 2 CONTENTS 1. Rationality of Groups 2. Centers of

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RATIONALITY OF GROUPS AND CENTERS OF INTEGRAL GROUP RINGS

Andreas Bächle Groups St Andrews 2017

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NOTATION. G finite group ZG integral group ring of G U(ZG) group of units of ZG

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CONTENTS

  • 1. Rationality of Groups
  • 2. Centers of Integral Group Rings
  • 3. Solvable Groups
  • 4. Frobenius Groups
  • 5. References
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CONTENTS

  • 1. Rationality of Groups
  • 2. Centers of Integral Group Rings
  • 3. Solvable Groups
  • 4. Frobenius Groups
  • 5. References
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DEFINITIONS. x ∈ G.

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DEFINITIONS. x ∈ G. x rational in G :⇔ ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x

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DEFINITIONS. x ∈ G. x rational in G :⇔ ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x x semi-rational in G :⇔ ∃ m ∈ Z ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x

  • r

xj ∼ xm

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DEFINITIONS. x ∈ G. x rational in G :⇔ ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x x semi-rational in G :⇔ ∃ m ∈ Z ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x

  • r

xj ∼ xm x inverse semi-rational in G :⇔ ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x or xj ∼ x−1

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DEFINITIONS. x ∈ G. x rational in G :⇔ ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x x semi-rational in G :⇔ ∃ m ∈ Z ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x

  • r

xj ∼ xm x inverse semi-rational in G :⇔ ∀ j ∈ Z

(j,o(x))=1

: xj ∼ x or xj ∼ x−1 G is called rational :⇔ ∀ x ∈ G: x is rational in G etc.

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For χ ∈ Irr(G), x ∈ G set Q(χ) := Q({χ(y): y ∈ G}) Q(x) := Q({ψ(x): ψ ∈ Irr(G)}).

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For χ ∈ Irr(G), x ∈ G set Q(χ) := Q({χ(y): y ∈ G}) Q(x) := Q({ψ(x): ψ ∈ Irr(G)}). G rational ⇔ CT(G) ∈ Qh×h

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For χ ∈ Irr(G), x ∈ G set Q(χ) := Q({χ(y): y ∈ G}) Q(x) := Q({ψ(x): ψ ∈ Irr(G)}). G rational ⇔ CT(G) ∈ Qh×h G semi-rational ⇔ ∀x ∈ G: [Q(x) : Q] ≤ 2

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For χ ∈ Irr(G), x ∈ G set Q(χ) := Q({χ(y): y ∈ G}) Q(x) := Q({ψ(x): ψ ∈ Irr(G)}). G rational ⇔ CT(G) ∈ Qh×h G semi-rational ⇔ ∀x ∈ G: [Q(x) : Q] ≤ 2 G inverse semi-rational ⇔ ∀x ∈ G: Q(x) ⊆ Q( √ −dx), dx ∈ Z≥0 ⇔ ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ), dχ ∈ Z≥0
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For χ ∈ Irr(G), x ∈ G set Q(χ) := Q({χ(y): y ∈ G}) Q(x) := Q({ψ(x): ψ ∈ Irr(G)}). G rational ⇔ CT(G) ∈ Qh×h G semi-rational ⇔ ∀x ∈ G: [Q(x) : Q] ≤ 2 G inverse semi-rational ⇔ ∀x ∈ G: Q(x) ⊆ Q( √ −dx), dx ∈ Z≥0 ⇔ ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ), dχ ∈ Z≥0

CT(G) =    ...   

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For χ ∈ Irr(G), x ∈ G set Q(χ) := Q({χ(y): y ∈ G}) Q(x) := Q({ψ(x): ψ ∈ Irr(G)}). G rational ⇔ CT(G) ∈ Qh×h G semi-rational ⇔ ∀x ∈ G: [Q(x) : Q] ≤ 2 G inverse semi-rational ⇔ ∀x ∈ G: Q(x) ⊆ Q( √ −dx), dx ∈ Z≥0 ⇔ ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ), dχ ∈ Z≥0

CT(G) =    ...    CT(G) =    ...    CT(G) =     ...    

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EXAMPLES.

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EXAMPLES.

◮ Sn is rational.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

P rational ⇔ p = 2

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

P rational ⇔ p = 2 and n ≤ 12.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24

  • r

p = 3 and n ≤ 18.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24

  • r

p = 3 and n ≤ 18. DEFINITION.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24

  • r

p = 3 and n ≤ 18. DEFINITION. π(G) = {p prime: p | |G|}, the prime spectrum of G.

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EXAMPLES.

◮ Sn is rational. ◮ P ∈ Sylp(Sn).

P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ {2, 3}.

◮ P ∈ Sylp(GL(n, pf)).

P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24

  • r

p = 3 and n ≤ 18. DEFINITION. π(G) = {p prime: p | |G|}, the prime spectrum of G. Then |π(Sn)| − → ∞ for n → ∞.

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G rational G inverse semi-rational G semi-rational

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G rational G inverse semi-rational G semi-rational G solvable

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G rational G inverse semi-rational G semi-rational = ⇒ G solvable π(G) ⊆ {2, 3, 5}

Gow, 1976

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

G simple

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

G simple 3 groups

Feit-Seitz, 1989

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

G simple 3 groups all An + 41 groups

Feit-Seitz, 1989 Alavi-Daneshkhah, 2016

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

G simple 3 groups 25 groups all An + 41 groups

Feit-Seitz, 1989 Alavi-Daneshkhah, 2016

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

G simple 3 groups 25 groups all An + 41 groups

Feit-Seitz, 1989 Alavi-Daneshkhah, 2016

|G| ≤ 511

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G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ ? = ⇒ ? G solvable π(G) ⊆ {2, 3, 5} π(G) ⊆ {2, 3, 5, 7, 13} π(G) ⊆ {2, 3, 5, 7, 13, 17}

Gow, 1976 Chillag-Dolfi, 2010

|G| odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational

Classified by Chillag-Dolfi, 2010

G simple 3 groups 25 groups all An + 41 groups

Feit-Seitz, 1989 Alavi-Daneshkhah, 2016

|G| ≤ 511 ≈ 1% ≈ 46% ≈ 61%

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CONTENTS

  • 1. Rationality of Groups
  • 2. Centers of Integral Group Rings
  • 3. Solvable Groups
  • 4. Frobenius Groups
  • 5. References
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U(ZG)

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±G ⊆ U(ZG) – “trivial units”

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±G ⊆ U(ZG) – “trivial units” ±G = U(ZG) ⇔ G abelian with exp G | 4 or exp G | 6

  • r

G Hamiltonian 2-group (Higman, 1940)

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±G ⊆ U(ZG) – “trivial units” ±G = U(ZG) ⇔ G abelian with exp G | 4 or exp G | 6

  • r

G Hamiltonian 2-group (Higman, 1940) Z(U(ZG))

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±G ⊆ U(ZG) – “trivial units” ±G = U(ZG) ⇔ G abelian with exp G | 4 or exp G | 6

  • r

G Hamiltonian 2-group (Higman, 1940) ±Z(G) ⊆ Z(U(ZG)) – “trivial central units”

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±G ⊆ U(ZG) – “trivial units” ±G = U(ZG) ⇔ G abelian with exp G | 4 or exp G | 6

  • r

G Hamiltonian 2-group (Higman, 1940) ±Z(G) ⊆ Z(U(ZG)) – “trivial central units” ±Z(G) = Z(U(ZG)) ⇔: G cut group (all central units trivial)

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±G ⊆ U(ZG) – “trivial units” ±G = U(ZG) ⇔ G abelian with exp G | 4 or exp G | 6

  • r

G Hamiltonian 2-group (Higman, 1940) ±Z(G) ⊆ Z(U(ZG)) – “trivial central units” ±Z(G) = Z(U(ZG)) ⇔: G cut group (all central units trivial)

  • U(ZG) :
  • (ZG)1, Z(U(ZG))
  • < ∞
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±G ⊆ U(ZG) – “trivial units” ±G = U(ZG) ⇔ G abelian with exp G | 4 or exp G | 6

  • r

G Hamiltonian 2-group (Higman, 1940) ±Z(G) ⊆ Z(U(ZG)) – “trivial central units” ±Z(G) = Z(U(ZG)) ⇔: G cut group (all central units trivial)

  • U(ZG) :
  • (ZG)1, Z(U(ZG))
  • < ∞

ր տ

  • ften up to f.i.

by “bicyclic units” covered by “bi- cyclic units” & “Bass units”

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THEOREM (Ritter-Sehgal, et.al.) For a finite group G TFAE (1) G is cut. (2) ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ),

dχ ∈ Z≥0.

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THEOREM (Ritter-Sehgal, et.al.) For a finite group G TFAE (1) G is cut. (2) ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ),

dχ ∈ Z≥0. (3) G is inverse semi-rational.

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THEOREM (Ritter-Sehgal, et.al.) For a finite group G TFAE (1) G is cut. (2) ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ),

dχ ∈ Z≥0. (3) G is inverse semi-rational. (4) K1(ZG) is finite.

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THEOREM (Ritter-Sehgal, et.al.) For a finite group G TFAE (1) G is cut. (2) ∀χ ∈ Irr(G): Q(χ) ⊆ Q(

  • −dχ),

dχ ∈ Z≥0. (3) G is inverse semi-rational. (4) K1(ZG) is finite. In particular: G cut ⇒ G/N cut for all N G.

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CONTENTS

  • 1. Rationality of Groups
  • 2. Centers of Integral Group Rings
  • 3. Solvable Groups
  • 4. Frobenius Groups
  • 5. References
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THEOREM (Bakshi-Maheshwary-Passi, 2016) G = 1 cut-group (1) 2 ∈ π(G) or 3 ∈ π(G). (2) If G is nilpotent, then G is a {2, 3}-group. (3) If G is metacyclic, then G is in a list of 52 groups.

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THEOREM (Bakshi-Maheshwary-Passi, 2016) G = 1 cut-group (1) 2 ∈ π(G) or 3 ∈ π(G). (2) If G is nilpotent, then G is a {2, 3}-group. (3) If G is metacyclic, then G is in a list of 52 groups. THEOREM (Maheshwary, 2016) Let G be a solvable cut group. (1) If |G| is odd = ⇒ π(G) ⊆ {3, 7} and all elements of G are of prime power order. (2) If |G| is even and all elements of G are of prime power order = ⇒ π(G) ⊆ {2, 3, 5, 7}.

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THEOREM (Bakshi-Maheshwary-Passi, 2016) G = 1 cut-group (1) 2 ∈ π(G) or 3 ∈ π(G). (2) If G is nilpotent, then G is a {2, 3}-group. (3) If G is metacyclic, then G is in a list of 52 groups. THEOREM (Maheshwary, 2016) Let G be a solvable cut group. (1) If |G| is odd = ⇒ π(G) ⊆ {3, 7} and all elements of G are of prime power order. (2) If |G| is even and all elements of G are of prime power order = ⇒ π(G) ⊆ {2, 3, 5, 7}. THEOREM (B., 2017) Let G be a solvable cut group. Then π(G) ⊆ {2, 3, 5, 7}.

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THEOREM (B., 2017) Let G be a solvable cut group. Then π(G) ⊆ {2, 3, 5, 7}.

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THEOREM (B., 2017) Let G be a solvable cut group. Then π(G) ⊆ {2, 3, 5, 7}. Strategy of proof.

◮ π(G) ⊆ {2, 3, 5, 7, 13}

(Chillag-Dolfi).

◮ Let G be a minimal counterexample, V G minimal. ◮ Then G ≃ V ⋊ G/V,

G/V is again cut.

◮ The F13[G/V]-module V has the “12-eigenvalue property”. ◮ Derive restrictions on field of character values of V. ◮ By a result of Farias e Soares such a module cannot exist

for a solvable group G/V.

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CONTENTS

  • 1. Rationality of Groups
  • 2. Centers of Integral Group Rings
  • 3. Solvable Groups
  • 4. Frobenius Groups
  • 5. References
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THEOREM (B., 2017). Let K be a Frobenius complement. (1) If |K| is even ... (2) If |K| is odd ...

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THEOREM (B., 2017). Let K be a Frobenius complement. (1) If |K| is even and the compelement of a cut Frobenius group G, then G is isomorphic to a group in the series on the left (b, c, d ∈ Z≥1) or one of the groups on the right. (a) Cb

3 ⋊ C2

(α) C2

5 ⋊ Q8

(b) C2b

3 ⋊ C4

(β) C2

5 ⋊ (C3 ⋊ C4)

(c) C2b

3 ⋊ Q8

(γ) C2

5 ⋊ SL(2, 3)

(d) Cc

5 ⋊ C4

(δ) C2

7 ⋊ SL(2, 3)

(e) Cd

7 ⋊ C6

(f) C2d

7 ⋊ (Q8 × C3)

Conversely, for each of the above structure descriptions, there is a unique cut Frobenius group. (2) If |K| is odd ...

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THEOREM (B., 2017). Let K be a Frobenius complement. (1) If |K| is even ... (2) If |K| is odd, then there is a cut Frobenius group G if and

  • nly if K ≃ C3 and the kernel F is a group admitting a

fixed-point free automorphism σ of order 3 such that

(a) F is a cut 2-group.

In particular, |F| = 22a, a ∈ Z≥1 and F is an extension of an abelian group of exponent a divisor of 4 by an an abelian group of exponent a divisor of 4.

(b) F is an extension of an elementary abelian 7-group by an elementary abelian 7-group, exp F = 7 and σ fixes each cyclic subgroup of F.

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Strategy of proof. G cut Frobenius group with complement K.

◮ K is also cut. ◮ Show that K is solvable, so π(G) ⊆ {2, 3, 5, 7}. ◮ Determine possible structures of P ∈ Sylp(K). ◮ Determine possible structures of K. ◮ Use irreducible representations of these complements to

describe structure of some G.

◮ Decide which subdirect products of the groups above are

cut Frobenius groups.

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REFERENCES

  • A. BÄCHLE, Integral group rings of solvable groups with trivial

central units, 2017, ❛r❳✐✈✿✶✼✵✶✳✵✹✸✹✼❬♠❛t❤✳●❘❪. G.K. BAKSHI, S. MAHESHWARY, I.B.S. PASSI, Integral group rings with all central units trivial, J. Pure Appl. Algebra, 221(8), 1955-1965, 2017, ❛r❳✐✈✿✶✻✵✻✳✵✻✽✻✵❬♠❛t❤✳❘❆❪.

  • D. CHILLAG, S. DOLFI, Semi-rational solvable groups, J. Group

Theory 13(4), 535-548, 2010.

  • S. MAHESHWARY, Integral group rings with all central units trivial:

solvable groups, 2016, ❛r❳✐✈✿✶✻✶✷✳✵✽✸✹✹❬♠❛t❤✳❘❆❪.

  • J. RITTER, S.K. SEHGAL, Integral group rings with trivial central

units, Proc. Amer. Math. Soc. 108(2), 327-329, 1990.