CERTAIN APPLICATIONS OF THE BURNSIDE RINGS AND

GHOST RINGS IN THE REPRESENTATION THEORY OF FINITE GROUPS (II)

K. K. NWABUEZE AND F. VAN OYSTAEYEN

Abstract. Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent A(G) of finite p-groups was obtained in [4]. In this paper, we compute A(G) for any finite group G – hence providing the global version of [4].

1. Introduction

This is a continuation of the paper [4], where we obtained a new setting for the Artin exponent of a finite group and described these exponents for finite p-groups. In this note we show that the restriction made in [4] that G is a p-group can be remove. In otherwords the Artin exponent for any finite group G depends on the value of a p-subgroup of G. We claim that the Artin exponent A(G , U) of a finite G is equal to 1 if and only if G is cyclic whereas A(G , U) difers by p from the order of a p-subgroup of G or is equal to the order of a p-subgroup of G if G is noncylic and theorem 4.2 dictates which. If the featuring p-subgroups of G is Dihedral, Quoternion or semi-dihedral, then A(G , U) is 2 or 4 and theorem 4.4 dictates which. In section 3 we show that we can reduce our problem essentially to the case where G is a finite p-groups. For the convenience of the reader, we recall in section 2 some results in Burnside ring theory and the useful results in [4]. Finally in section 4, we described the Artin exponent for finite group G.

Some notation: G in this paper denotes a finite group. Let 1_{G} denote the trivial subgroup of G; p, a prime number; < T >, subgroup of G generated by T; < T >_{N}, the normal subgroup of G generated by T; g_{h}, the g conjugate of h; [g , h], the commutator of g and h − in particular [S , T] :=< [s , t] | s ∈ S, t ∈ T > . Let G^{sub/∼} denote a set of representatives of the conjugacy classes of subgroups of G and Z(G) denote the center of G. Let G_{s} := {g ∈ G | gs = s ∀ s ∈ S}, the stabilizer subgroup of s.

Dr. Nwabueze’s Research at MSRI is supported in part by the University of Antwerpen (UIA)

and the United States National Science Foundation grant DMS-9022140.

2 K. K. NWABUEZE AND F. VAN OYSTAEYEN

2. Preliminaries

Let G be a finite group. The set of isomorphism classes of finite left G-sets form a commutative semi ring B^{+}(G) with respect to disjoint union and cartesian product. Using the Grothendieck ring construction on B^{+}(G) one obtains the corresponding Grothendieck ring B(G) associated with B^{+}(G). This resulting ring is called the Burnside ring B(G) of G. So the Burnside ring is the Grothendieck ring of finite G-sets which is generated as an algebra over Z by the isomorphism classes of finite left G-sets S_{1} S_{2} … subject to the relations S_{1}−S_{2} = 0 if S_{1} ≅ S_{2}; S_{1}+S_{2}−(S_{1}⊔Y) = 0 and S_{1}·S_{2}−(S_{1}×S_{2}) = 0. In otherwords, the elements of the Burnside ring are the virtual G-sets, that is the formal difference S_{1} − S_{2} of isomorphism classes of finite G-sets S_{1}, S_{2}. Furthermore one has that S_{1}−S_{2} = S′_{1}−S′_{2} ⟺ S_{3}⊔S_{1}⊔S′_{2} ≅ S_{3}⊔S_{2}⊔S′_{1} where S′_{1}, S′_{2} and S_{3} are G-sets. This of course is equivalent with S_{1} ⊔ S′_{2} ≅ S_{2} ⊔ S′_{1}. For any subgroup U ≤ G of G the map χ_{U} : S ⟶ #{s ∈ S | U ≤ G_{s}}, from S onto the number of U invariant elements in S, induces a homomorphism from B(G) into Z – the integers. Using standard arguments concerning the Burnside ring the following statements are easily verified (see [2]).

(a) All homomorphism from B(G) into Z is of the form χ_{U} defined above.

(b) For any subgroup U, V ≤ G one has that χ_{U} = χ_{V} if and only if U ∼ V (U is G-conjugate to V).

(c) The product map χ := Π_{i=1}^{k} χ_{i} : B(G) ⟶ Π_{i=1}^{k} Z of k different homomorphisms is

injective and maps B(G) onto a subring of finite index Π_{i=1}^{k}(N_{G}(U_{i} : U_{i}) of Π_{i=1}^{k} Z –

this way identifying Π_{i=1}^{k} Z with the integral closure B̃(G) of B(G) in its total quotient

ring Q ⊗_{z} B(G) ≅ Π_{i=1}^{k} Q (see [2],[5]). The product Π_{i=1}^{k} Z is called the Ghost ring of G.

It is well known (see [2]) that there are canonical set of congruence relation where by one can characterize the image χ(B(G)) as a subgroup of Π_{i=1}^{k} Z. More exactly, if U ⊴ V ≤ G then for a G-set X the Burnside lemma applied with respect to the V/U-set X^{U}, implies that ∑_{i=i}^{k} χ_{<\overline{v}>}(X) ≡ 0(mod (V : U)), where < \overline{v} > denotes the subgroup of V ≤ G generated by the coset \overline{v} ⊆ V.

Lemma 2.1. Identifying B(G) with its image in B̃(G) with respect to the map

Π_{U∈G^{sub/∼}} χ_{U} one has that {n ∈ Z | nB̃(G) ⊆ B(G)} = |G|Z.

CERTAIN APPLICATIONS OF THE BURNSIDE RINGS 3

Because for U, V ≤ G, one has χ_{U} = χ_{V} if and only if U and V are G-conjugate and for x, y ∈ B(G), one has χ_{U}(x) = χ_{U}(y) for all U ≤ G if and only if x = y it follows that we can identify each x ∈ B(G) with the associated map (also denoted by x) U ⟶ χ_{U}(x), from the set of all subgroups of G into Z. Therefore we can consider B(G) in a canonical way as a subring of the ghost ring B̃(G). For more detail on the Burnside rings see [2]. We recall the following simple facts (see any standard book on group theory).

Definition 2.2. Let G be a 2-group with generators g and h. We say that: (A) G is quaternion (Q) if g, h have the relations

g^{2^{n}} = 1, h^{2} = g^{2^{n−1}}, hgh^{−1} = g^{−1}.

(B) G is dihedral (D) if g, h have the relations

g^{2^{n}} = 1, h^{2} = 1, hgh^{−1} = g^{−1}.

(C) G is semi-dihedral (SD) if g, h have the relations

g^{2^{n}} = 1, h^{2} = 1, hgh^{−1} = g^{−1+2^{n−1}}.

Lemma 2.3. Let G be a noncyclic abelian p-group (p ≠ 2) and U ⊴G, |U| = p. Then there exists a normal subgroup V ⊴ G of G of order p^{2}, containing U and isomorphic to C_{p} × C_{p}.

Lemma 2.4. Let G be a finite p-group and U a subgroup of G of order less than p^{β}. For g ∈ G we have that < g , U > is cyclic of order p^{β} if and only if U ⊆< g > .

Lemma 2.5. Let G be a finite p-group and let U and H be subgroups of G such that U is a cyclic normal subgroup with U ⊆ H. Let C(H) denote the set of all cyclic subgroups V of H with U ≤ V and (V : U) = p, and let C′(H) = C′_{U}(H) denote the subset of C(H), consisting of those V ∈ C(H) which are normal in H. We have: (a.) |C(H)| ≡ |C′(H)| (mod p). (b.) H′ = HU^{′} := {h ∈ H | [h , H] ⊆ U} is a normal subgroup of H. (c.) C′(H) = C(H′) (d.) If G′ ⊆ H ⊴ G then |C(H)| = |C(G′)|(mod p).

Lemma 2.6. With the above notation, one has that if |U| = p ≠ 2 and [G , G] ⊆ U, then the following are eqivalent:

(i) |C(G)| = 1 (ii) |C(G)| ≢ 0(mod p) (iii) G is cyclic

4 K. K. NWABUEZE AND F. VAN OYSTAEYEN

Lemma 2.7. If G is abelian, then the following are equivalent:

(i) |C(G)| = 1 (ii) |C(G)| ≢ 0(mod p) (iii) G is cyclic

Lemma 2.8. If |U| = p ≠ 2, then the following are eqivalent:

(i) |C′(G)| = 1 (ii) |C′(G)| ≢ 0(mod p) (iii) G′ is cyclic

Lemma 2.9. With the definitions and assumptions of (3.1), if G is a 2-group, one has (a.) |C(H)| ≡ 1(2) ⇒ Z(G) is cyclic. (b.) If [G , G] ⊆ U and |C(H)| is odd, then G is cyclic or nonabelian of order 8. (c.) If H ≤ G with [H , H] ⊆ U ⊆ H and |{V ∈ C(H) | V ⊆ H}| ≡ 1(mod 2), then H is cyclic or nonabelian of order 8. (d.) If |C(H)| ≡ 1(mod 2), then either H′ is cyclic or G is nonabelian of order 8. (e.) If |C(H)| ≡ 1(mod 2), then G contains a cyclic normal subgroup of index 2.

In [4], we characterized the elements in B̃(G) which are in B(G) in the following way:

Definition and Theorem 2.10. For U a family of cyclic subgroups of G. Define e_{U} ∈ B̃(G) by e_{U}(U) = 1 if U is contained in U and 0 otherwise. Then one has that |G|·e_{U}(U) ∈ B(G). Furthermore define A(G , U) := min(n ∈ N | n·e_{U} ∈ B(G)). The integer A(G , U) is said to be the Artin exponent of G. One also has that A(G , U) divides |G| (see [4]).

In [4] we also gave a Burnside ring theoretic proof of the following well known results of Lam (see [3]).

Theorem 2.11. Let G be a finite p-group. One has that A(G , U) = 1 if and only if G is cyclic.

Theorem 2.12. Let G be a noncyclic finite p-group (p ≠ 2) of order p^{α}. Then one has A(G , U) = p^{α−1}.

Theorem 2.13. If p = 2 and G is a noncyclic 2-group of order 2^{α} then A(G) = 2^{α−1}, excluding the cases where G is the Quaternion or Dihedral in which cases one has that A(G , U) = 2.

These results provides the complete computation of A(G , U) where G is a p-group.

CERTAIN APPLICATIONS OF THE BURNSIDE RINGS 5

3. Reduction to p-Groups

Let p be a prime and consider Z_{p}, the localization of Z at p, that is Z_{p} := {a/b | a ∈ Z, b ∈ Z − pZ} ⊆ Q. Put B_{p}(G) := Z_{p} ⊗_{z} B(G), and B̃_{p}(G) := Z_{p} ⊗_{z} B̃(G). One has that Z_{p} · χ(B(G)) ≅ B_{p}(G) := Z_{p} ⊗_{z} B(G) coincides with the subgroup of B̃_{p}(G) for which the relations ∑ n·e_{U} ≡ 0(mod (V_{p} : U)), with V_{p} representing a subgroup of G between U and V for which V_{p}/U is the Sylow p-subgroup of V/U (see [2]). Therefore e_{U} is characterized as a subgroup of Π_{i=1}^{k} Z by those relations, which runs through all the prime divisors of |G|. Hence we have;

Lemma 3.1. An element n·e_{U} ∈ B̃(G) is contained in B(G) if and only if for every U ⊴ V ≤ G with (V : U) a prime power one has ∑_{vU∈V/U} n · e_{U}(< v , U >) ≡

0(mod (V : U)), where e_{U}(< v , U >) := #{vU ∈ V/U | < v , U > ∈ U}.

Corollary 3.2. For U ⊴ V ≤ G, assume that (V : U) = p^{α} for some prime p. Consider the decomposition U = U_{p} × U_{p′} where |U_{p}| = p^{β} is a power of p and |U_{p′}| is prime to p. Let V_{p} denote the Sylow subgroup of V. Then one has

(a) (V : U) = p^{α} implies that V = V_{p} ∝ U_{p′} and V = V_{p} × U_{p′} if and only if U ≤ Z(V), the center of V. (Here ∝, resp. × denotes the semidirect product, resp. direct product).

(b) (V : U) = (V_{p} : U_{p}).

(c) If c(U , V) := #{vU ∈ V/U | < v , U > is cyclic } and c(U_{p} , V_{p}) := #{vU_{p} ∈ V_{p}/U_{p} | < v , U_{p} > is cyclic } then c(U , V) = c(U_{p} , V_{p}) if V = V_{p} × U_{p′}, that is U ≤ Z(V).

Remark 3.3. Observe that for vU ∈ V/U the group < v , U > is cyclic only if v ∈ C_{V} (U) := {w ∈ V | wu = uw for all u ∈ U}. This implies that (V : C_{V} (U)) must always divide the number ”n” in order for (V : U) to divide n · c(U , V). So we have the following strategy for computing the Artin exponent. For each p subgroup V of G, consider the set of all cyclic normal subgroups U of V with U contained in the centre of V and compute the minimum of those positive integers n for which (V : U) divides n · c(U , V). Finally, it is easy to see that if U ≤ Z(V), for a p-group V such that U cyclic one has that, c(U , V) = c(U/U^{p} , V/U^{p}), where U^{p} denotes the unique subgroup of U of index p. So we can restrict ourselves to the case where |U| = p.

We now summarize with the following, Theorem 3.4. Let G be a finite group. Then A(G , U) = A(G_{p} , U), where G_{p} is a p-subgroup of G.

4. Global Discussion

For this discussion it suffices to consider the following cases;

\underline{Case 1:} G is cyclic.

The first thing to note is that the proof of Theorem(2.8) in [4] does not depend on the fact that G is a p-group. For the sake of explicitness we state the following global version of the theorem. That is we remove the condition that G is a p-group.

Theorem 4.1. Let G be a finite group. For n ∈ N and a subgroup U ≤ G, let x_{n} ∈ B̃(G) be defined by

x_{n} = \begin{cases}n, & if U is cyclic \\ 0, & otherwise \end{cases}

Then A(G , U) = 1 if and only if G is cyclic.

Proof: We check by congruences. Assume U ⊴ V ≤ G such that U cyclic of order p and U is contained in the center of V, where |V | = p^{α}, (p a prime). Then x ∈ B(G)

implies ∑_{vU∈V/U} x(< v , U >) = n · #{vU ∈ V/U | < v , U > is cyclic}. To compute

A(G , U), we let V = G, that is |G| = p^{α}. These assumption remains valid for the balance of this paper. Now if n = 1, then one has that #{gU ∈ G/U | < g , U > is cyclic } ≡ 0(mod (G : U)). So (G : U) | #{gU ∈ G/U | < g , U > is cyclic }. This implies |G| | #{g ∈ G | < g , U > is cyclic }. So for all g ∈ G, one has that < g , U > is cyclic. This implies that G is cyclic since if G is noncyclic then from (2.3) there exists a normal subgroup W ⊴ G such that W contains U and W ≅ C_{p} × C_{p} ≅ U × C_{p} = U× < g > for some g ∈ G. That is < g , U > = W for some g ∈ G. So G is cyclic.

The converse is obvious. □

\underline{Case 2:} G is noncylic.

Theorem 4.2. With the assumptions of Theorem 4.1 one has that if G is noncyclic then

A(G , U) := \begin{cases}p^{α}, & if G′ := \{g ∈ G | [g , U] ⊆ U\} is cyclic \\ p^{α−1}, & if otherwise. \end{cases}

Proof: Since G is noncyclic, then one obviously has that |G| does not divide #{g ∈ G | < g , U > is cyclic}. If G is abelian recall from lemma 2.3 that there exist a normal subgroup W of order p^{2} of the form {W ⊴ G | U ≤ W ≅ C_{p} × C_{p}} and indeed we have (see lemma 2.5 and 2.7) that C(W) := #{C ≤ W | C is cyclic C ⊇ U} ≡ 0(mod p). This implies A(G , U) = p^{α−1}.

If G be nonabelian and G′ is noncyclic then repeating the above argument (by setting G = G′, one has that A(G , U) = p^{α−1}. If G′ is cyclic then it is easy to derive (from 2.8) that #{g ∈ G | < g , U > is cyclic } ≢ 0(mod p), in which case A(G , U) = p^{α}. □

We now compute A(G , U) for the special groups Q, D, and SD.

Corollary 4.3. Keeping the assumptions of (4.1) one has that if G is any of Q, D, or SD then

A(G , U) := \begin{cases}4, & if G′ is cyclic \\ 2, & if otherwise. \end{cases}

Proof: Using the notation in section 3 one has here that if C(G) ≡ 0(mod 2) then by a repetition of the above arguments one has that A(G , U) = 2^{α−1}. If C(G) ≢ 0(mod 2) then, from lemma 2.9, G is cyclic or a nonabelian group of order 8. But by our assumption G is noncyclic and so G must be a nonabelian group of order 8. It is now easy to see (by simple computation using lemma 2.9 repeatedly) that A(G) = 4 when G′ is cyclic and A(G) = 2 otherwise. □

References

[1] A. DRESS; A characterization of solvable groups, Math. Z. 110, 1960, pp. 213-217.

[2] A. DRESS; Notes on the theory of representation of finite groups, Lecture Notes, Bielefeld, 1971 [3] T.Y. LAM; Artin exponent for finite groups, J. Algebra 9, (1968), pp. 94-119.

[4] K.K. NWABUEZE; Certain applications of the Burnside ring and its ghost ring in finite group theory I Bull. Soc. Math. Belgium (to appear).

[5] T. TOM-DIECK; Transformation groups and representation theory, Lecture Notes in Math. 444, Springer-Verlag (1975).

[6] K. YAMAUCHI; On the 2-part of Artin exponent of finite groups, Sci. Rep. Tokyo Kyoiku Daigaku, sect. A 10, (1970), pp 234 - 240.

K. K. Nwabueze, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, U.S.A.

E-mail address: nwabueze@msri.org

F. Van Oystaeyen, Department of Maths and Computer Science, University of Antwerpen (UIA), 2610 Wilrijk, BELGIUM

E-mail address: voyst@uia.ua.ac.be