Torsion Units in Integral Group Rings

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Leo Margolis University of Stuttgart (With A. Bächle) Jahrestagung DFG Schwerpunkt 1489 Osnabrück September 28th - October 2nd 2015

Notation and Setting G finite group ZG integral group ring over G Augmentation map: ε : ZG Z, ε( g G z g g) = z g g G V (ZG) group of units of augmentation 1, aka normalized units All units of ZG are of the form ±V (ZG), so it suffices to condider V (ZG)

Questions How can one construct generic units in V (ZG) and generate big subgroups in V (ZG)? E. Jespers, Á. del Río: Group Ring Groups, De Gruyter 2015. General Question: How is the torsion part of V (ZG) connected to the group base G? E.g.: Is every finite subgroup of V (ZG) isomorphic to a subgroup of G? (No, Hertweck 97) Do the orders of torsion elements of V (ZG) and G coincide? (Unknown. Yes, if G is solvable (Hertweck 07).) In general we know: X V (ZG) finite, then X divides G (Zhmud-Kurennoi 67). Moreover exp(v (ZG)) = exp(g) (Cohn-Livingstone 65).

Questions How can one construct generic units in V (ZG) and generate big subgroups in V (ZG)? E. Jespers, Á. del Río: Group Ring Groups, De Gruyter 2015. General Question: How is the torsion part of V (ZG) connected to the group base G? E.g.: Is every finite subgroup of V (ZG) isomorphic to a subgroup of G? (No, Hertweck 97) Do the orders of torsion elements of V (ZG) and G coincide? (Unknown. Yes, if G is solvable (Hertweck 07).) In general we know: X V (ZG) finite, then X divides G (Zhmud-Kurennoi 67). Moreover exp(v (ZG)) = exp(g) (Cohn-Livingstone 65).

Zassenhaus Conjecture and Prime Graph Question Main open question concerning torsion units: (First) Zassenhaus Conjecture (H.J. Zassenhaus, in the 60s) (ZC) For u V (ZG) of finite order there exist an unit x QG and g G s.t. x 1 ux = g. If such x and g exist, one says that u is rationally conjugate to g. Prime Graph Question (Kimmerle, 2006) (PQ) Let p and q be different primes. If V (ZG) contains an element of order pq, does G contain an element of order pq?

Zassenhaus Conjecture and Prime Graph Question Main open question concerning torsion units: (First) Zassenhaus Conjecture (H.J. Zassenhaus, in the 60s) (ZC) For u V (ZG) of finite order there exist an unit x QG and g G s.t. x 1 ux = g. If such x and g exist, one says that u is rationally conjugate to g. Prime Graph Question (Kimmerle, 2006) (PQ) Let p and q be different primes. If V (ZG) contains an element of order pq, does G contain an element of order pq?

Some known results on ZC The Zassenhaus Conjecture is known to hold for Abelian Groups (Higman 39) Nilpotent Groups (Weiss 91) Groups with normal Sylow subgroup with abelian complement (Hertweck 07) Metacyclic Groups (Hertweck 08) Cyclic-By-Abelian Groups (Caicedo-M -del Rio 13) Two other special series of metabelian groups (Sehgal-Weiss 86, Marciniak-Ritter-Sehgal-Weiss 87) Groups till order 71 (Höfert 04) PSL(2, q) for q 25 (Luthar-Passi, Hertweck, Gildea, Kimmerle-Konovalov, Bächle-M ) Some other specific non-solvable, non-simple groups

Some known results on ZC The Zassenhaus Conjecture is known to hold for Abelian Groups (Higman 39) Nilpotent Groups (Weiss 91) Groups with normal Sylow subgroup with abelian complement (Hertweck 07) Metacyclic Groups (Hertweck 08) Cyclic-By-Abelian Groups (Caicedo-M -del Rio 13) Two other special series of metabelian groups (Sehgal-Weiss 86, Marciniak-Ritter-Sehgal-Weiss 87) Groups till order 71 (Höfert 04) PSL(2, q) for q 25 (Luthar-Passi, Hertweck, Gildea, Kimmerle-Konovalov, Bächle-M ) Some other specific non-solvable, non-simple groups

Some known results on PQ Theorem (Kimmerle-Konovalov, 2012) Suppose that (PQ) has an affirmative answer for each almost simple image of G, then it has also a positive answer for G. (G almost simple, if S G Aut(S) for some non-ab. simple S.) Keeping this in mind (PQ) is known for: Solvable groups (Kimmerle 06) Groups whose order is divisible by at most three different primes (Kimmerle-Konovalov 12, Bächle-M 14) Almost simple groups whose socle is In the first half of the sporadic simple groups (Bovdi-Konovalov et al. 07-12) PSL(2, p), with p a prime (Hertweck 08, M 14) An alternating group of degree up to 16 (Luthar-Passi, Hertweck, Salim, Bächle-Caicedo 15) Most of these results rely on the so-called HeLP-Method.

Some known results on PQ Theorem (Kimmerle-Konovalov, 2012) Suppose that (PQ) has an affirmative answer for each almost simple image of G, then it has also a positive answer for G. (G almost simple, if S G Aut(S) for some non-ab. simple S.) Keeping this in mind (PQ) is known for: Solvable groups (Kimmerle 06) Groups whose order is divisible by at most three different primes (Kimmerle-Konovalov 12, Bächle-M 14) Almost simple groups whose socle is In the first half of the sporadic simple groups (Bovdi-Konovalov et al. 07-12) PSL(2, p), with p a prime (Hertweck 08, M 14) An alternating group of degree up to 16 (Luthar-Passi, Hertweck, Salim, Bächle-Caicedo 15) Most of these results rely on the so-called HeLP-Method.

HeLP-Package The HeLP-method, named after Hertweck, Luthar and Passi (name due to A. Konovalov) uses ordinary and modular characters to tackle these questions. It has now been implemented into a GAP-package, available at http://homepages.vub.ac.be/abachle/help/ Two main purposes: Reserachers working in the field can use it to obtain new results. Results using the method can be checked by readers.

HeLP-Method: Partial Augmentations Let x G be a conjugacy class in G and u = ε x (u) = g x G z g g G z g g ZG. Then is called the partial augmentation of u with respect to x. Theorem (Marciniak-Ritter-Sehgal-Weiss 87) u V (ZG) is rationally conjugate to an element of G if and only if ε x (u d ) 0 for all x G and divisors d of n. Theorem (Higman 39, Berman 53) If u V (ZG) is a torsion unit, then ε 1 (u) = 0 or u = 1. Theorem (Hertweck 07) Let u be a torsion unit in V (ZG) and x G s.t. ε x (u) 0. Then the order of x divides the order of u.

HeLP-Method: Partial Augmentations Let x G be a conjugacy class in G and u = ε x (u) = g x G z g g G z g g ZG. Then is called the partial augmentation of u with respect to x. Theorem (Marciniak-Ritter-Sehgal-Weiss 87) u V (ZG) is rationally conjugate to an element of G if and only if ε x (u d ) 0 for all x G and divisors d of n. Theorem (Higman 39, Berman 53) If u V (ZG) is a torsion unit, then ε 1 (u) = 0 or u = 1. Theorem (Hertweck 07) Let u be a torsion unit in V (ZG) and x G s.t. ε x (u) 0. Then the order of x divides the order of u.

Example: u V (ZA 5 ), o(u) = 2 5 (Luthar-Passi, 1989) 1a 2a 3a 5a 5b χ 4 0 1 1 1, Z(χ) = Z. o(u 5 ) = 2, χ(u 5 ) = ε 2a (u 5 )χ(2a) = 0 D(u 5 ) diag(1, 1, 1, 1) o(u 6 ) = 5, χ(u 6 ) = ε 5a (u 6 )χ(5a) + ε 5b (u 6 )χ(5b) = (ε 5a (u 6 ) + ε 5b (u 6 )) ( 1) = 1 D(u 6 ) diag(ζ, ζ 2, ζ 3, ζ 4 ), ζ 5 = 1 ζ

Example: u V (ZA 5 ), o(u) = 2 5 (Luthar-Passi, 1989) 1a 2a 3a 5a 5b χ 4 0 1 1 1, Z(χ) = Z. o(u 5 ) = 2, χ(u 5 ) = ε 2a (u 5 )χ(2a) = 0 D(u 5 ) diag(1, 1, 1, 1) o(u 6 ) = 5, χ(u 6 ) = ε 5a (u 6 )χ(5a) + ε 5b (u 6 )χ(5b) = (ε 5a (u 6 ) + ε 5b (u 6 )) ( 1) = 1 D(u 6 ) diag(ζ, ζ 2, ζ 3, ζ 4 ), ζ 5 = 1 ζ u = u 5 u 6 D(u) D(u 5 ) D(u 6 ) and χ(u) / Z : Contradiction!

Example: u V (ZA 5 ), o(u) = 2 5 (Luthar-Passi, 1989) 1a 2a 3a 5a 5b χ 4 0 1 1 1, Z(χ) = Z. o(u 5 ) = 2, χ(u 5 ) = ε 2a (u 5 )χ(2a) = 0 D(u 5 ) diag(1, 1, 1, 1) o(u 6 ) = 5, χ(u 6 ) = ε 5a (u 6 )χ(5a) + ε 5b (u 6 )χ(5b) = (ε 5a (u 6 ) + ε 5b (u 6 )) ( 1) = 1 D(u 6 ) diag(ζ, ζ 2, ζ 3, ζ 4 ), ζ 5 = 1 ζ u = u 5 u 6 D(u) D(u 5 ) D(u 6 ) and χ(u) / Z : Contradiction!

The HeLP-Inequalities Theorem (Luthar-Passi 89; Hertweck 04) Let u ZG torsion unit of order n, F splitting field for G with p = char(f ) n, χ a (p-brauer) character of an 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 n Tr Q(ζ d )/Q(χ(u d )ζ dl ) Z 0. d n

The HeLP-Inequalities This yields a system of inequalities for the partial augmentations ε x (u) of u, assuming knowledge on the partial augmentations of the powers u d for divisors d of the order of u: 1 n = 1 n Tr Q(ζ d )/Q(χ(u d )ζ dl ) d n d n d 1 Tr Q(ζ d )/Q(χ(u d )ξ dl ) + 1 n x G : x is p regular ε x (u)tr Q(ζ)/Q (χ(x)ξ l ) In a less technical way: For every (p-brauer) character χ of G and character ψ of u we have χ u, ψ u Z 0.

The HeLP-Inequalities This yields a system of inequalities for the partial augmentations ε x (u) of u, assuming knowledge on the partial augmentations of the powers u d for divisors d of the order of u: 1 n = 1 n Tr Q(ζ d )/Q(χ(u d )ζ dl ) d n d n d 1 Tr Q(ζ d )/Q(χ(u d )ξ dl ) + 1 n x G : x is p regular ε x (u)tr Q(ζ)/Q (χ(x)ξ l ) In a less technical way: For every (p-brauer) character χ of G and character ψ of u we have χ u, ψ u Z 0.

Example: u V (ZA 5 ), o(u) = 2 5 (Luthar-Passi, 1989) In the previous example, u V (ZA 5 ), o(u) = 2 5, this yields: µ 0 (u, χ, 0) = 2 /5 (ε 5a (u) + ε 5b (u)) µ 1 (u, χ, 0) = 1 /10 (ε 5a (u) + ε 5b (u)) + 1 /2 µ 2 (u, χ, 0) = 1 /10 (ε 5a (u) + ε 5b (u)) + 1 /2 µ 5 (u, χ, 0) = 2 /5 (ε 5a (u) + ε 5b (u)) As it is easily seen, the above expressions can not be all non-negative integers.

Generic Results For series of groups possesing generic character tables computing motivating examples can lead to generic results. E.g.: For G = PSL(2, p f ) and r p a prime, elements of r-power order in V (ZG) are rationally conjugate to elements of G. (M 14) For G = PSL(2, p) and r p an odd prime, there is exactly one non-trivial possibility for partial augmentations of elements of order 2r satisfying the HeLP-constraints. (del Río-Serrano 15) Some generic results for alternating and symmetric groups were recently obtained by Bächle-Caicedo.

Application: Groups of small order (Krauß-Meyer-Ritter) Group Order GAP-Id and result of HeLP-method investigating ZC 72 15, 22, 23, 24, 31, 33, 35, 40(6), 42, 43, 44 96 13, 30, 31, 40, 41, 43, 64, 65(8), 66, 67(8), 76, 77, 81-83, 85, 86, 89-93, 95-97, 101-105, 131, 136, 137, 142-147, 151, 153, 154, 160, 185(4), 186(4), 187(4), 188-193, 194(4), 195(4), 226, 227(2,4) 120 5, 35, 37, 38, 39 144 31, 32, 33(4), 54-62, 64-67, 86-89, 91, 93-100, 109, 115(6,12), 116, 117(6), 118(12), 119(12), 120, 121, 122, 123(4,12), 124, 125, 126(4), 127-129, 131, 133, 135-142, 144, 145, 147, 148, 150-154, 168, 170-175, 177, 182(6), 183, 186(6), 187, 188-190 160 13, 30, 31, 40, 41, 43, 74, 80, 86, 88, 90, 91, 95-97, 99, 100, 103-105, 106, 107, 109-111, 115-119, 145, 150, 151, 156, 157, 158-160, 161, 165, 167, 168, 174, 234(2,4) 168 23, 43(6), 45, 46, 48, 49, 53 180 17, 19, 22, 24, 25, 27, 29, 30, 36 192* 180(4,8), 181(4,8), 182(4), 183(8), 184, 185(4), 944-954, 955(2,4), 956, 957(4), 958(4,8), 959(4,8), 960(4,8), 961(4), 962(8), 963(8), 964(8), 965(8), 966(8), 967(8), 968(8), 969(4), 970(4), 971(4), 972(4), 973(4,8), 974(4,8), 975(4,8), 976(4,8), 977-980, 981(8), 982(8), 983-986, 987(4,8), 988(8),989(8), 990(4,8), 991(4), 1468(4), 1469(4), 1470(4), 1471(4), 1472(4), 1473(4), 1474-1476, 1477(4), 1478(4), 1479-1486, 1487(4), 1488(4), 1495(2,4), 1538(2,4) (* not all necessary groups tested) 200 24, 25, 26, 32, 34, 36, 43(10)

Application: PQ for 4-primary groups The simple groups with four different prime divisors: PSL(2, p) A 7 PSL(3, 4) PSU(3, 8) P Ω + (8, 2) PSL(2, 2 f ) A 8 PSL(3, 5) PSU(3, 9) Sz(8) PSL(2, 3 f ) A 9 PSL(3, 7) PSU(4, 3) Sz(32) A 10 PSL(3, 8) PSU(5, 2) G 2 (3) PSL(2, 16) PSL(3, 17) PSp(4, 4) 3 D 4 (2) PSL(2, 25) PSL(4, 3) PSp(4, 5) 2 F 4 (2) PSL(2, 27) PSU(3, 4) PSp(4, 7) M 11 PSL(2, 49) PSU(3, 5) PSp(4, 9) M 12 PSL(2, 81) PSU(3, 7) PSp(6, 2) J 2 HeLP-method Lattice-method No method yet

How much is possible? There are some natural bounds for the package: Not all known characters are available in the GAP-Library yet. For this reason our package allows using any class function, entered e.g. manually or obtained by inducing of characters from subgroups. The main computations always concern solving the inequalities. So far we use for that purpose: The zsolve-function from 4ti2 (Walter) and the 4ti2-Interface for GAP (Gutsche) to solve the inequalities. The redund-function from lrs (Avis) to reduce the systems to smaller sizes. (Its behaviour with zsolve is sometimes strange.) Can we solve some inequalities unsovable so far, due to runtime, using an other solver? Maybe Normaliz?

How much is possible? There are some natural bounds for the package: Not all known characters are available in the GAP-Library yet. For this reason our package allows using any class function, entered e.g. manually or obtained by inducing of characters from subgroups. The main computations always concern solving the inequalities. So far we use for that purpose: The zsolve-function from 4ti2 (Walter) and the 4ti2-Interface for GAP (Gutsche) to solve the inequalities. The redund-function from lrs (Avis) to reduce the systems to smaller sizes. (Its behaviour with zsolve is sometimes strange.) Can we solve some inequalities unsovable so far, due to runtime, using an other solver? Maybe Normaliz?

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