RATIONALITY OF GROUPS AND CENTERS OF INTEGRAL GROUP RINGS Andreas - PowerPoint PPT Presentation

RATIONALITY OF GROUPS AND CENTERS OF INTEGRAL GROUP RINGS Andreas Bächle 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 Integral Group Rings 3. Solvable Groups 4. Frobenius Groups 5. References 3

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

x ∈ G . DEFINITIONS . 5

x ∈ G . DEFINITIONS . : x j ∼ x : ⇔ ∀ j ∈ Z x rational in G ( j , o ( x ))= 1 5

x ∈ G . DEFINITIONS . : x j ∼ x : ⇔ ∀ j ∈ Z x rational in G ( j , o ( x ))= 1 : x j ∼ x x j ∼ x m : ⇔ ∃ m ∈ Z ∀ j ∈ Z x semi-rational in G or ( j , o ( x ))= 1 5

x ∈ G . DEFINITIONS . : x j ∼ x : ⇔ ∀ j ∈ Z x rational in G ( j , o ( x ))= 1 : x j ∼ x x j ∼ x m : ⇔ ∃ m ∈ Z ∀ j ∈ Z x semi-rational in G or ( j , o ( x ))= 1 : x j ∼ x or x j ∼ x − 1 x inverse semi-rational in G : ⇔ ∀ j ∈ Z ( j , o ( x ))= 1 5

x ∈ G . DEFINITIONS . : x j ∼ x : ⇔ ∀ j ∈ Z x rational in G ( j , o ( x ))= 1 : x j ∼ x x j ∼ x m : ⇔ ∃ m ∈ Z ∀ j ∈ Z x semi-rational in G or ( j , o ( x ))= 1 : x j ∼ x or x j ∼ x − 1 x inverse semi-rational in G : ⇔ ∀ j ∈ Z ( j , o ( x ))= 1 : ⇔ ∀ x ∈ G : x is rational in G G is called rational etc. 5

For χ ∈ Irr ( G ) , x ∈ G set Q ( χ ) := Q ( { χ ( y ): y ∈ G } ) Q ( x ) := Q ( { ψ ( x ): ψ ∈ Irr ( G ) } ) . 6

For χ ∈ Irr ( G ) , x ∈ G set Q ( χ ) := Q ( { χ ( y ): y ∈ G } ) Q ( x ) := Q ( { ψ ( x ): ψ ∈ Irr ( G ) } ) . CT ( G ) ∈ Q h × h ⇔ G rational 6

For χ ∈ Irr ( G ) , x ∈ G set Q ( χ ) := Q ( { χ ( y ): y ∈ G } ) Q ( x ) := Q ( { ψ ( x ): ψ ∈ Irr ( G ) } ) . CT ( G ) ∈ Q h × h ⇔ G rational ⇔ ∀ x ∈ G : [ Q ( x ) : Q ] ≤ 2 G semi-rational 6

For χ ∈ Irr ( G ) , x ∈ G set Q ( χ ) := Q ( { χ ( y ): y ∈ G } ) Q ( x ) := Q ( { ψ ( x ): ψ ∈ Irr ( G ) } ) . CT ( G ) ∈ Q h × h ⇔ G rational ⇔ ∀ x ∈ G : [ Q ( x ) : Q ] ≤ 2 G semi-rational √ ⇔ ∀ x ∈ G : Q ( x ) ⊆ Q ( − d x ) , d x ∈ Z ≥ 0 G inverse semi-rational � ⇔ ∀ χ ∈ Irr ( G ) : Q ( χ ) ⊆ Q ( − d χ ) , d χ ∈ Z ≥ 0 6

For χ ∈ Irr ( G ) , x ∈ G set Q ( χ ) := Q ( { χ ( y ): y ∈ G } ) Q ( x ) := Q ( { ψ ( x ): ψ ∈ Irr ( G ) } ) . CT ( G ) ∈ Q h × h ⇔ G rational ⇔ ∀ x ∈ G : [ Q ( x ) : Q ] ≤ 2 G semi-rational √ ⇔ ∀ x ∈ G : Q ( x ) ⊆ Q ( − d x ) , d x ∈ Z ≥ 0 G inverse semi-rational � ⇔ ∀ χ ∈ Irr ( G ) : Q ( χ ) ⊆ Q ( − d χ ) , d χ ∈ Z ≥ 0   ... CT ( G ) =     6

For χ ∈ Irr ( G ) , x ∈ G set Q ( χ ) := Q ( { χ ( y ): y ∈ G } ) Q ( x ) := Q ( { ψ ( x ): ψ ∈ Irr ( G ) } ) . CT ( G ) ∈ Q h × h ⇔ G rational ⇔ ∀ x ∈ G : [ Q ( x ) : Q ] ≤ 2 G semi-rational √ ⇔ ∀ x ∈ G : Q ( x ) ⊆ Q ( − d x ) , d x ∈ Z ≥ 0 G inverse semi-rational � ⇔ ∀ χ ∈ Irr ( G ) : Q ( χ ) ⊆ Q ( − d χ ) , d χ ∈ Z ≥ 0       ... ...   CT ( G ) = CT ( G ) = CT ( G ) =       ...       6

EXAMPLES . 7

EXAMPLES . ◮ S n is rational. 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . P rational ⇔ p = 2 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . P rational ⇔ p = 2 and n ≤ 12. 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24 p = 3 and n ≤ 18. or 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24 p = 3 and n ≤ 18. or DEFINITION . 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24 p = 3 and n ≤ 18. or DEFINITION . π ( G ) = { p prime : p | | G |} , the prime spectrum of G . 7

EXAMPLES . ◮ S n is rational. ◮ P ∈ Syl p ( S n ) . P rational ⇔ p = 2. P inverse semi-rational ⇔ p ∈ { 2 , 3 } . ◮ P ∈ Syl p ( GL ( n , p f )) . P rational ⇔ p = 2 and n ≤ 12. P inverse semi-rational ⇒ p = 2 and n ≤ 24 p = 3 and n ≤ 18. or DEFINITION . π ( G ) = { p prime : p | | G |} , the prime spectrum of G . Then | π ( S n ) | − → ∞ for n → ∞ . 7

G rational G inverse semi-rational G semi-rational 8

G rational G inverse semi-rational G semi-rational G solvable 8

G rational G inverse semi-rational G semi-rational = ⇒ π ( G ) ⊆ { 2 , 3 , 5 } G solvable Gow, 1976 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable Gow, 1976 Chillag-Dolfi, 2010 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable Gow, 1976 Chillag-Dolfi, 2010 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable Gow, 1976 Chillag-Dolfi, 2010 | G | odd 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable Gow, 1976 Chillag-Dolfi, 2010 | G | odd G = 1 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable Gow, 1976 Chillag-Dolfi, 2010 | G | odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational Classified by Chillag-Dolfi, 2010 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable Gow, 1976 Chillag-Dolfi, 2010 | G | odd G = 1 G semi-rational ⇐ ⇒ G inverse semi-rational Classified by Chillag-Dolfi, 2010 G simple 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable 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 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable 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 A n + 41 groups Feit-Seitz, 1989 Alavi-Daneshkhah, 2016 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable 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 A n + 41 groups Feit-Seitz, 1989 Alavi-Daneshkhah, 2016 8

G rational G inverse semi-rational G semi-rational = ⇒ = ⇒ = ⇒ ? ? π ( G ) ⊆ { 2 , 3 , 5 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 } π ( G ) ⊆ { 2 , 3 , 5 , 7 , 13 , 17 } G solvable 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 A n + 41 groups Feit-Seitz, 1989 Alavi-Daneshkhah, 2016 | G | ≤ 511 8