ABOUT THE Q-GROUPS WITH MAXIMAL AUTOMORPHISM

PROPERTY

ION ARMEANU AND DIDEM OZTURK

Abstract. A Q-group G is a finite group with all characters rational valued.

This is equivalent with Aut(K) ' NG(K) / CG(K) for all cyclic subgroups ofG. In this note we will completely classify the Q-groups G with this maximal

property for Aut(K) not only for cyclic subgroups, but also for the Sylow

subgroups of G.

All groups will be finite and we shall use the notations and definitions of [2] .

Definition 1. A finite group G is a Q-group iff every character of G is rationalvalued.

Proposition 1. A group G is a Q-group if and only if for every x, y G withx = y there is a g G such that gxg1 = y.Proposition 2. A group G is a Q-group if and only if for all x G,NG(x)/CG(x) ' Aut(x).

Proof. Let f : NG(x) Aut(x) defined by f(g) = gxg1. Then f is a ho-momorphism, Ker f = CG (x) and f is in. It is clear that f is onto iff G is aQ-group.

Remark 1. G is a Q-group iff Aut(K) ' NG(K) / CG(K) for all cyclic subgroupsK of G.

Theorem 1. Let G be a Q-group. If Aut(K) ' NG(K)/CG(K) for the 2-Sylowsubgroups K of G then the 2-Sylow subgroups of G are isomorphic with Z2 , thecommutator subgroup G

is a 3-group, G ' GZ2 and Z2 inverts all elements of

G.

Proof. Let S be a 2-Sylow subgroup of G. Suppose that S has order 2n, n 2. ByGaschutz theorem ( [1] pg. 403 Satz 19.1), 2/ | A(S)/I(S) | where A(S) = Aut(S)and I(S) are the inner authormorphisms of S. Since Aut(S) ' NG(S)/CG(S) thereis an element x / S of order a power of 2 and x NG(S). This contradicts themaximality of S. Hence n = 1 and S ' Z2.

By Walters theorem [3] G has a normal subgroup N O2 (G) with G/N of oddorder and N/O2 (G) ' S P where S is a 2-group and P is a direct product ofsimple groups of the form L2(k), k 3 , k 3, 5mod(8) , or k = 2p , or the Jankosimple group J(11), or is of Ree type.

Since G is a Q-group, N = G and G/O2 (G) ' S P .

1991 Mathematics Subject Classification. 20C15, 20C11 .Key words and phrases. group theory, characters, Q-groups.

1

2 ION ARMEANU AND DIDEM OZTURK

Because the simple groups listed before are not Q-groups, it follows that G/O2 (G) 'S. Therefore G is 2-nilpotent and S Syl2(G). For every Q-group, O2 (G) G

(see [2]), therefore G

= O2 (G).

We prove now that G

is a 3-group. We use induction on | G |. Let L be aminimal normal subgroup of G. Clearly L is an elementary abelian p-group.

If p = 2, then L ' S ' Z2 Syl2(G) is normal in G, hence L = G ' Z2.If p 6= 2, then G/L is a non-trivial Q-group and by induction | G/L |= 2 3a pb.

We show that p = 3. Let x G of order p and X = x. Then, NG(X) S Syl2(G) and NG(X) CG(X) CG(X)S NG(X). Since CG(X)S isselfnormalising in G, it follows that CG(X)S = NG(X). Then NG(x)/CG(X) 'S/(S CG(X)) ' Z2 and thus p = 3. Corollary 1. Let G be a Q-group and Aut(K) ' NG(K)/CG(K) for all 2-Sylowsubgroups K of G , then G ' S1, S2, S3. (Sn the symmetric groups)Proof. Anologously with the proof of theorem 1 for p = 2, it follows that the orderof a 3-Sylow subgroup is 3. Corollary 2. Let G be a Q-group with all Sylow subgroups cyclic. Then G 'S1, S2, S3.

Proof. Aut(K) ' NG(K)/CG(K) for all Sylow subgroups K. Corollary 3. Let G be a Q-group with all proper subgroups cyclic. Then G 'S1, S2, S3.

References

[1] [1] B.Huppert, Endliche Gruppen, Springer-Verlag, 1967.

[2] D.Kletzing, Structure and Representations of Q-groups, Lecture Notes in Mathematics,Springer-Verlag, 1984.

[3] J.H.Walter, The characterisation of finite groups with abelian Sylow 2-groups, Annals of Math.

89 (1969), 405-514

University of Bucharest, Faculty of Physics, Department of Theoretical Physics

and Mathematics, 077125, Magurele, P.O.Box MG-11, BUCHAREST ROMANIA

E-mail address: ion.armeanu@gmail.com

Mimar Sinan Guzel Sanatlar Universitesi Mimar Sinan Fine Arts University ISTAN-

BUL TURKIEE-mail address: dislekel@yahoo.com