The Artin exponent induced from cyclic subgroups of finite groups was studied
extensively by T.Y. Lam. A Burnside ring theoretic version of Lam's results for
$p$-groups was given by the author in an earlier paper. Here we look at the
Artin exponent induced from the elementary abelian subgroups of finite
$p$-groups using some results of A. Dress

Nwabueze, K. K.

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arXiv

Nwabueze, Kenneth K.

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The Artin exponent of finite groups

Institute of Mathematics AS CR, v. v. i.

Acta Mathematica et Informatica Universitatis OstraviensisKenneth K. NwabuezeThe Artin exponent of finite groupsActa Mathematica et Informatica Universitatis Ostraviensis, Vol. 5 (1997), No. 1, 71--79Persistent URL: http://dml.cz/dmlcz/120516Terms of use:© University of Ostrava, 1997Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czActa Mathematica et Informatica Universitatis Ostraviensis 5 (1997) 71-79 71 The Artin Exponent of Finite Groups* Kenneth K. Nwabueze 1. Introduction The Artin exponent induced from cyclic subgroups of finite groups was studied extensively by T.Y. Lam in [5]. A new setting for the description of these exponents in the sense of [5] using the Burnside ring theoretic methods was given in [6]. In this paper we are interested in defining a similar concept for elementary abelian subgroups of finite groups. More precisely, using basically some results of A. Dress concerning the Burnside rings (see[l], [2], [3]), we shall prove the following main results (see chapter three for the explanation of the notation). Main result: Let G be a finite p-group and e(G) the Artin exponent induced from the elementary abelian subgroups of G. One has the following: (a)If G is abelian, then e(G) = \G : U|, where U is the maximal elementary abelian subgroup of G. (b) If G is a quaternion or dihedral group then e(G) = 2 and e(G) = 4 if G is the semidihedral group. (c) In all other cases e(G) = \G\/p. The above results seem to suggest that the invariant e(G) gives an interesting numerical measure of the deviation of G from being an elementary abelian group. Let me mention where this study fits in. First, the method we are going to adopt here is Burnside ring theoretic and hence affirms the utility of the Burnside ring in the representation theory of finite groups. Second, the ideas of the Artin exponent appeals to number theory - which in themselves have aesthetic values. Finally, exponents of this type have been known to be very useful in the formulation of some induction theorems for various functors on the category of finite groups- see for instance [8]. *Reserch at MSRI is supported by USA National Science Foundation grant DMS-9022140 72 Kenneth K. Nwabueze This paper is arranged as follows. In section 2, we collect some well known definitions and basic results about the Burnside rings. In section 3, we form our definition and provide some particular examples. In section 4, we prove some useful results from which the main result finally follows in section 5. 2. The Rings ft(G) and Q(G) Let G be a finite group. A G-set is a finite set on which G acts from the left by permutation. The set of isomorphism classes of finite left G-sets form a commu-tative semi-ring 0 + ( G ) with addition induced by disjoint union and multipli-cation induced by cartesian product with diagonal actions. The Grothendieck ring of 0 + ( G ) is denoted by 0(G) and called the Burnside ring of G. Let Gsub/~ denote the set of conjugate classes of subgroups of G. For any G Xu := {x £ X | ux = x \/u € U}, be the set of U-invariant elements of X. Since the map |X| —> \XU\ preserves sums and products, it extends to a ring homomorphism Xu : ft(G) —> Z : [X] —» \XU\ from (1(G) to Z (the integers). Now for every (U) E Gsub/~ let Z^u) be the ring isomorphic to Z. We define the ghost ring 0(G) as follows: 0(G) := n %(U)- S° 0(G) is a ring by pointwise (Ll)GGsu6/~ multiplication. It is well known (see [3]) that the product map X~ n X(U) : (1(G) - + 0(G) (U)EG*»b/~ is injective. One also has the following important result. Theorem 2.1. Identifying 0(G) with its image in 0(G) with respect to the map U X(U) one has that {n <E Z \ n • 0(G) C 0(G)} = \G\ • Z. (U)€G'*b/~ Proof: see [3] Lemma 2.2. For every x € 0(G) one has Y^ X<g>(x) = O(mod |Gj), where geG < g > denotes the cyclic group generated by g. (This relation is called the Cauchy-Frobenius-Burnside relation). The Artin Exponent of Finite Groups 73 Proof: see [4] Because for any two subgroups U and V in G one has that xu — Xv if and only if U~V (U and V are conjugate in G) and for x and y in ft(G) one has Xu(x) — Xu(y) for all subgroups U in G if and only if x — y it follows that we can identify each x £ Q(G) with the associated map U —>• xu(x), from the set of all subgroups of G into Z. We shall denote this associated map also by x. So we ca Q(G). Theorem 2.3. Letx G 0(G). Then x G 0(G) i/and Only if for every U<V <G with (V : U) a power of a prime one has ]T x(< v , U >) ~ 0(raOd (V : vuev/u U)). Proof: see [2] For more details on the Burnside ring see [3], [7]. 3. The Artin Exponent We now give the definition of the Artin exponent in terms of the Burnside ring. This arises in the following context. Definition 3.1. Let XL denote the family of all elementary abelian subgroups of G. Let bu be the element of Cl(G) defined as follows. If (U) is the conjugacy class of the subgroup U of G then Ьu(U) : = { J | UЄU U£U We call the integer e(G) := min(n € At | n • bu € fl(G)) the Artin exponent ofG. Corollary 3.2. \G\ • bu(U) e 0(G). 74 Kenneth K. Nwabueze Proof: Follows from Theorem 2.1. Corollary 3.3. e(G) divides the order of G. Proof: Put e := e(G) and b := bu(U). Write \G\ = qe + r with 0 < r < e. Now r • b = (\G\ ~qe)-b = \G\ - b - (qe) • b G 0(G). Hence r = 0 in view of the minimality of e. To motivate the general algebraic procedure to be followed in the coming sections we take some particular examples. Example 3.4. For a finite group G, let 8(G) denote the set of all subgroups of G. Now let G be a cyclic p-group of order p1. That is G =< g >, \G\ = p1 and $(G) = {U0, Ui, U2, • • , U 7 - i } , where U0 = < gpl >; Ui = < g^"1 > ; . . . . ; U7_t = < gpl > and \Uj\ = pi. The family U of elementary abelian subgroups of G consists of the subgroups Uo and U\. From definition 3.1, we have that e(G) • bu = e(G) • (1, 1, 0, . . . ,0) = (e(G), e(G), 0, . . . ,0) € Q(G). To finish this example we need the following lemma. Lemma 3.5. Let G denote a cyclic p-group of order pn. For every i G {0, . . ,n} let Ui denote the unique subgroup of order p1 of G. Then given x(Ui) G (l(G) n one has that x(U{) G Q(G) if and only ifpl-x(Ui) + ]>_ (p3 -p?~l) -x(Uj) = j=i+l 0(modpn), where \Uj\ = pP. Proof: For x G fi(G) and U < V < G we have that J2 x(< v , U >) = 0 (mod V : U). vuev/u Since G is abelian, Ui < G Vi G {0, . . . , n } . Hence by Theorem 2.3 ] T Xi(< g , U, >) = Oimodp"-1). gUieG/Ux The Artin Exponent of Finite Groups 75 Since x(< g , Ui >) = x(Uj) if and only if < o , Uz > = Uj we have £ x ( < o , U 2 > ) = E ^ - ) ' # W € G/U, | < a , U,>= Uj }. gU^G/Ui j=i Since we have that < <? , Ui > = < g > if G ^ U?; then JT x(U)3 • #{gUt e G/Ui I < g , Ut > = U,} - x(ft) + E j W i X(UJ) • ̂ ^ = 0(mod if*-*) <=> x(Uz) • p* + E ; = i + 1 ^(U,) • (p* - p ^1 ) = 0(mod p»). Now we finish our example by observing from the above lemma that for i = 1 we have e(G) • p = 0(mod p7) and for i = 0 we have e(G) + e ( G ) - ( p - l ) = 0(mOd p7) 4=> p 7 " 1 | e(G). This implies e(G) = p 7 " 1 . Example 3.6. Consider U a family of elementary abelian subgroups of a finite group G. Then, from lemma 3.5, one has that e(G) — 1 if and only ifU — 8(G). In particular, if G is elementary abelian then e(G) = 1. Example 3.7. Let G be the quaternion group of order 8. Then it is to see by direct computation that e(G) = 2. We now generalize the above examples in the following section. 4. Results We start this section with the following observation. Lemma 4 .1. For U a family of elementary abelian subgroups of a finite group G let eu(G) be the Artin exponent with respect to the family U. Assume that eu(G) = 1, then for all U, V < G with U <3 V and (V : U) a prime power, one has that U G U if and only if V G IX. 76 Kenneth K. Nwabueze Proof: Assume (V : U) = pa for some a > 0. Then x G 0(G) implies ~*(U) = x(V)(mod p). Therefore U e U implies x(V) = l(mOd p), hence x(V) / 0. We have x(V) = 1 and so V G U. Conversely U ^ U implies x(V) = 0(mod p), hence ~*(V) ^ 1 which implies x(V) = 0, that is V $ U. Lemma 4.2. If G is solvable, then x G 0(G) implies that U = S(G) Or U = 0 (rTie empty set). Proof: If G is solvable then for every U in G there exists a sequence of subgroups U = Uo, Ui, . . . . , Uk = 1 such that U; is normal in Uj_i for i = 1, . . . . . . k and (U;_i : Ui) is a prime power or 1. Hence if U ^ 0, then by (4.1) one has that 1 G U and U G U for all U < G. Corollary 4.3. If G is a finite p-group, then e(G) — 1 if and only if U = S(G). Proof: Follows from 4.2. Proposition 4.4. Let G be a finite noncyclic abelian p-group. Then one has e(G) — (G : U), where U = {g G G | op = 1} w the maximal elementary abelian subgroup of G. Proof: Let U be the family of all elementary abelian subgroups of G and let U G U. Let e(G) and bu be defined as in 3.1. By definition of e(G) we have e(G) • bu G O(G), hence, by Theorem 2.3, £ e(G) • bu(< g , U >) = pLlGG/Cf Q(mod \G : U|). Since < a , U > is elementary abelian if and only if a G U, we have 53 M<9 ,U>) gUeG/U = |{#U G G/U | < 5 , U > is elementary abelian }| = \{gU 6 G/U | < £ , U > < U}| The Artin Exponent of Finite Groups 77 = |U : U\. Hence it follows that e(G) is the minimal positive integer such that e(G)-\U : U| = 0(mod\G : U|), that is e(G) = (G : U). Before we state the next result we need the following (see [9]). Lemma 4.5. Let G be a non-abelian p-group of order pn(p = 2). If there exists a g G G of order pn~~l then G has one of the following presentations for some h in G. (A) g2n = 1, li2 = a2" , hgh~l = a - 1 , where p = 2 and n > 3, - the quaternion group. (B) g2 = 1, h2 = 1, hgh~l = a - 1 , where p = 2 and n > 3 - the dihedral group. (C) g2n~l = 1. h2 = 1. hgh~l = g1+2n~\ where p = 2 and n > 4. (D) a2""1 = 1. h2 = 1, hgh~l = g~l+2n~2. nj/jgre p = 2 and n > 4, - £/ze semi-dihedral group. Proposition 4.6. Let G be a finite nonabelian and noncyclic p group. Then one has that e(G) = \G\/p unless G is the quaternion, the dihedral or the semi-dihedral in which cases e(G) = 2 for the quaternion or the dihedral groups and e(G) = 4 for the semi-dehidral. Proof: As usual let U be the family of all elementary abelian subgroups of G and let [ / G i l . Let e(G) and bu be defined as before. So e(G) • bu € -1(G) and J2 e{G) - bu(< g,U>) = 0(mod \G : U|). gUeG/U To find e(G), it suffices to set |U| = p. Now Y, M<<7, ^ >) 5LIGG/CI 78 Kenneth K. Nwabueze = |{gU £ G/U | < g , U > is elementary abelian }| - 1 + \{gU e G/U | < O , U > =* Cp x C p } | , where Cp denote a cyclic group of order p. Let >f(0 , U) := |{gc/ e G/U | < O , U > £ Cp x C p } | . For G a noncyclic p-group, it is easy to see that N(g , U) = Q(raOci p) if G is not a quaternion or dihedral. This implies that e(G) = \G\/p is the desired minimal positive integer such that e(G) • (l-f-K(o , U)) = 0(mod (G : U)), that is e(G) — \G\/p. On the other hand if JV(o , U) ^ O(raOdp) then G corresponds to the nonabelian group of order 8 is just the noncyclic abelian group of order 8. So for the semi-dihedral e(G) = 4. 5. Proof of main result Finally we have; Theorem 5.1. Let G be a finite p-group. We have the following. (a)If G is abelian, then the Artin exponent e(G) = \G : U|, where U is the maximal elementary abelian subgroup of G. (b) If G is a quaternion or dihedral group, then e(G) = 2, and e(G) = 4 if G is a semidihedral group. (c) In all other cases e(G) = \G\/p Proof: Results now follows from proposition 4.4 and proposition 4.6. References [1] A. DRESS; A characterization of solvable groups, Math. Z. 110, (1960), pp. 213-217. [2] A. DRESS; Congгuence relations characterizing the representation ring of the Symmetric group, J. Algebra 2, (1986), pp 350 - 363. [3] A. DRESS; Notes on the theory of representation of finite groups, Lecture Notes, Bielefeld, 1971. The Artin Exponent of Finite Groups 79 [4] A. DRESS, C. SIEBENEICHER, T. YOSHIDA; An application of Burnside rings in elementary finite group theory, J. Algebra 1, (1992), pp 27 - 44-[5] T.Y. LAM; Artin exponent for finite groups, J. Algebra 9, (1968), pp. 94-119. [6] K.K. NWABUEZE; Certain applications of the Burnside ring and its ghost ring in finite group theory I Bull. Soc. Math. Belgium (to appear). [7] T. TOM-DIECK; Transformation groups and representation theory, Lecture Notes in Math. 444? Springer-Verlag 1975. [8] R. SWAN; Induced representation and projective modules, Ann. Math 71, (1960), pp 552 - 578. [9] H. ZASSENHAUS; Theory of Groups, Chelsea, NewYork, 1958. Address: Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Bandar Seri Begawan, BRUNEI DARUSSALAM. 

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Some definition of the Artin exponent of finite groups

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Institute of Mathematics AS CR, v. v. i.