Subgroups containing the centralizers of its nontrivial elements Commemorating Maria Silvia Lucido

Transcription

1 Subgroups containing the centralizers of its nontrivial elements Commemorating Maria Silvia Lucido Zvi Arad, Netanya Academic College and Wolfgang Herfort, TU Vienna Padova, March 27th, 2010

2 CC-subgroup, Examples CC-subgroups and the Prime Graph Towards a Classification The Classification Theorem Profinite and Locally Finite Groups Totally disconnected topologically locally finite groups

3 CC-subgroup, Examples Definition A proper subgroup M of a finite group G is a CC-subgroup provided that for every 1 m M its centralizer C G (M) is contained in M. Example Frobenius group (Frobenius kernel and Frobenius complement are CC-subgroups) Projective linear groups PSL(2, q) where q = 2 n and n 2 Suzuki groups Sz(q) where q = 2 2n+1 and n 1

4 CC-subgroups and the Prime Graph M is a π-hall subgroup of G the prime graph is not connected. Definition The prime graph has vertices the primes dividing the order of G and edges from p to q for primes p q if G contains an element of order pq. (O.H.Kegel & K.W.Gruenberg as has been noted in [10]) Therefore, for classifying the simple groups with a CC-group, one can use the tables by J.S. Williams, N.Yiori & H.Yamaki, A.S.Kondrat iev.

5 M.Suzuki, W.Feit & J.G.Thompson, M.Herzog, Z. Arad, Z.Arad & D.Chillag, S.Dolfi & E.Jabara & M.S.Lucido. For M of even order the above list of example in (2) is complete. (M. Suzuki [9]). For M of odd order and π := π(m) the following holds: Any π-hall subgroups of G are conjugate (Z.Arad & D.Chillag [1, 2]) If M is not nilpotent then NG (M) = M and, by [2], M is a Frobenius group. The prime graph has at least 3 components If M is nilpotent and N G (M) = M then G is Frobenius with complement M. (M.Herzog [7]) If M belongs to a prime component then it is nilpotent. (J.S.Williams [10]) If NG (M) = M and G is not Frobenius then G must be simple and the prime graph has at least 3 components (Z.Arad & W.Herfort [4], using a result by Z.Arad & M.Herzog [6])

6 The Classification Theorem Complete classification for M an odd CC-subgroup of G not a Frobenius group (Z.Arad & W.Herfort [4]). 1. M is non-nilpotent, G = PSL(2, q), q 3 (mod 4) and M is solvable of order M = q q 1 2 ; 2. M is nilpotent and one of the following holds: 2.1 G is simple non-abelian and G and M are classified in the tables contained in [4]; 2.2 G is not simple; putting H := (M) G, G/H and F (G) = F (H) = O π1 (H) are π 1 groups, S := H/F (H) is a simple group having the CC-subgroup MF (H)/F (H) = M and the pair (S,M) is of type (2.1); For G almost simple see M.S.Lucido [8]. 2.3 G is a 2-Frobenius group;

7 Profinite and Locally Finite Groups When G is locally finite then a local system of finite subgroups each containing a CC-subgroup exists. Z.Arad & W.Herfort classified in [3] all locally finite groups with a CC-subgroup that contains an involution. The pro-2 completion of the infinite dihedral group C 2 C 2 contains each free factor C 2 as a CC -subgroup. Restriction: M should be a Hall-subgroup. Then profinite Frobenius groups have been studied by D.Gildenhuys & W.Herfort & L.Ribes. A sample result from [5]: M must be either finite or open in G and G is the projective limit of finite groups with a CC-subgroup.

8 Totally disconnected topologically locally finite groups By definition such a group is locally compact and every compact subset is contained in a profinite subgroup. All profinite and locally finite groups belong to this class. Frobenius group: Q p C 2 with C 2 acting by inversion and p 2. Sample results [5]: If G is neither a Frobenius group nor a 2-Frobenius group then M must be locally finite. When M contains an involution then G is locally finite.

9 Z. Arad and D. Chillag, On Finite Groups Containing a CC-Subgroup, Arch.D.Math (1977) Z. Arad and D. Chillag, On Finite Groups Containing a CC-Subgroup, Arch.D.Math (1980) Z. Arad and W. Herfort, A classification of locally finite and profinite groups with a centralizer condition, Comm.Alg (1982) Z. Arad and W. Herfort Classification of Finite Groups with a CC-subgroup Comm.Alg. 32(6) (2004) Z. Arad and W. Herfort with a CC-subgroup, J.Lie Theory (2005)

10 Z. Arad and M. Herzog, A classification of groups with a centralizer condition II, Bull.Austr.Math.Soc (1977) Corrigendum and addendum, ibidem (1977) M. Herzog, On finite groups which contain a Frobenius group, J.Alg 6, , (1967) M.S. Lucido, Prime graph components of finite almost simple groups Rend.Semin.Mat.Univ.Padova 102, 1-22 (1999) Addendum ibidem 107, 1-2 (2002) M. Suzuki, On a class of doubly transitive groups, Annals of Math. II (1962) J.S. Williams, Prime Graph Components of Finite Groups, J.Alg (1981)