Extended Abstracts of the 41thIranian International Conference on Mathematics 12-15 September 2010, University of Urmia, Urmia, Iran, pp 212-214
ARTIN EXPONENT OF SOME RATIONAL GROUPS
H. SHARIFI∗
Abstract. A finite group whose irreducible complex characters are rational valued is called a rational group. We prove that if a rational group G contains a CC-subgroup or G is a direct prod- uct of Frobenius rational groups, then every rationally represented character is a generalized permutation character.
1. Introduction and Preliminaries
Let CharQ(G) and P(G) denote the ring of Z-linear combination of rationally represented characters and permutation characters of a finite group G, respectively. It is easy to see that P(G) a is subring of CharQ(G). By a theorem of Artin, | G | χ ∈ P(G) for all χ ∈ CharQ(G). The minimal number d ∈ N such that dχ ∈ P(G) for all χ ∈ CharQ(G), is called the Artin exponent of G and is denoted by γ(G). Indeed γ(G) is the exponent of CharQ(G)/P(G). The Artin exponent induced from cyclic subgroups of finite groups was studied extensively by T. Y. Lam in [2]. He proved that A(G) = exponent (PChar(G)Q(G)
cyclic)=1 if and only ifGis cyclic. One can show thatγ(G) divides theA(G) and therefore divides|G|. There is a fundamental distinction between γ(G) andA(G). While groups satisfying A(G) = 1 have been
2000Mathematics Subject Classification. Primary 20C15.
Key words and phrases. Rational group, Artin exponent, CC-subgroup, Local splitting.
∗ Speaker.
1
2 H. SHARIFI
characterized, there is no such characterization for groups satisfying γ(G) = 1. In this paper, we characterize all rational groups having a CC-subgroup, then prove that γ(G) = 1 for such rational groups. We also prove that if G is a direct product of Frobenius rational groups, then γ(G) = 1.
Definition 1.1. Let G be a finite group and pa prime number. Sup- pose that K is a Q-p-class of G and s is a p-regular element of K.
The group G is said to be locally split onK if CG(< s >)p is a direct factor ofNG(< s >)p; that is,NG(< s >)p =CG(< s >)p×V for some V ≤NG(< s >)p . The groupGis locally split at pifG is locally split on every Q-p-class K. The group G is everywhere locally split if G is locally split at all primes.
A finite group G is called a rational group or a Q-group, if all irre- ducible complex characters ofGare rational. For example, the symmet- ric groupSn and the Weyl groups of the classical complex Lie algebras are rational groups (for more details see [5]). A comprehensive descrip- tion of rational groups can be found in [1] and [6]. Throughout the paper, we use the following notations and terminology: H : K stands for the semi-direct product of the groupsH andK. Zndenotes a cyclic group of order n. E(pn) where p is a prime number, denotes the ele- mentary abelian p-group of orderpn and Q8 is employed to denote the quaternion group of order 8.
2. Main results
Theorem 2.1. ( [6], Corollary 212 A) If G is rational group which is everywhere locally split, then γ(G) = 1.
Definition 2.2. A proper subgroup H of G is called a CC-subgroup of G if CG(x)≤H for any x∈H−1.
Theorem 2.3. Let H be a CC-subgroup of odd order in a rational group G. Then H ∼=E(3n) or E(5n). Also
(i) G∼=E(3n) :Z2 , (ii) G∼=E(32m) :Q8
ARTIN EXPONENT OF SOME GROUPS 3
(iii) G∼=E(52) :Q8
Theorem 2.4. Let G be a rational group with a CC-subgroup, then γ(G) = 1.
Corollary 2.5. LetGis a rational group that having abelian 2-subgroup.
Then γ(G) = 1.
Theorem 2.6. Let G be a finite group that the number of kernels is equal to the number of classes. Then γ(G/O2(G)) = 1.
Remark 2.7. G. Segal, J. Ritter and J. Rasmussen in 1972 indepen- dently showed that γ(G) = 1 for any p-group G by using methods of algebraic K-theory [7]. Ritter and Rasmussen proved the same result using group-theoretic methods and tried to extend the result to nilpo- tent groups. However, J. Serre [3] remarks that γ(Z3×Q8) = 2, thus showed that the extension of Segal’s result to nilpotent groups required some modification. Rasmussen did this by showing that γ(G) = 1 or 2 when Gis nilpotent and providing necessary and sufficient condition for when each case occurs [4].
References
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2. T. L. Lam,Artin exponent for finite groups, J. Algebra9, (1968), 94-119.
3. J. P. Serre,Representations lineaires des groupes finis, Hermann, 1971.
4. J. Rasmussen,Rationally represented characters and permutation characters of nilpotent groups, J. Algebra29, (1974), 504- 509.
5. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson,Atlas of finite groups. Oxford University press, 1985.
6. D. Kletzing,Structure and representations ofQ-group, Lecture Notes in Math.
1084, Springer-Verlag, 1984.
7. G. Segal,Permutation representations of finite p-groups, Quart. J. Math. Ox- ford23, 1972, 375-381.
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Department of Mathematics Faculty of Science University of Sha- hed, Tehran, Iran.
E-mail address: [email protected]
