APPLICATIONS OF BURNSIDE RINGS IN ELEMENTARY GROUP THEORY

BY

Andreas DRESS, Christian SIEBENEICHER AND Tomoyuki YOSHIDA

Abstract: This is a report on some of the results which appear in [DSY 90]. A canonical ring homomorphism from the Burnside ring Ω(C) of a finite cyclic group C into the Burnside ring Ω(G) of any finite group G of the same order is exhibited and it is shown that many results from elementary finite group theory, in particular those claiming certain congruence relations, are simple consequences of the existence of this map.

Theorem: Let G be a finite group and let C denote the cyclic group of the same order n. There exists a ring homomorphism

α = α(G) : Ω(C) ⟶ Ω(G)

from the Burnside ring Ω(C) of the cyclic group C into the Burnside ring Ω(G) of the group G with the following property:

• for every subgroup U ≤ G of G and every element x ∈ Ω(C) one has

φ_{U}(α(x)) = φ_{C_{|U|}}(x)

where φ_{U}(α(x)) denotes the number of U–invariant elements in the virtual G–set α(x) and C_{|U|} denotes the unique subgroup of order |U| in C.

Remark: This theorem gives a precise conceptual interpretation of the observation ([Fr 95], [Hu 67], [Wa 80]) that quite a few elementary, but important results in group theory can be proved by comparing systematically certain group theoretic invariants of a finite group G with the same invariants for the cyclic group C of the same order.

The Burnside ring of a finite group For a finite group G we define its Burnside ring Ω(G) to be the Grothendieck ring of finite G–sets. It is generated as an algebra over Z by the (isomorphism classes of ) finite (left) G–sets X, Y, …, subject to the following relations:

X − Y = 0  if X ≅ Y ,
X + Y − (X ˙∪ Y) = 0,
X · Y − (X × Y) = 0.

The elements of Ω(G) are the virtual G–sets, i. e. , the formal differences X − Y of (isomorphism classes of) finite G–sets X, Y. Burnside numbers and the associated canonical homomorphisms Consider now for any G–set X and every subgroup U of the group G the subset

X^{U} := {x ∈ X | u · x = x for all u ∈ U}

of U–invariant elements of X. The mapping

X ↦ φ_{U}(X) := # X^{U}

which associates to the G–set X the number of U–invariant elements of X (its Burnside number with respect to the subgroup U) extends to a canonical ring homomorphism

φ_{U} : Ω(G) ⟶ Z.

Note that in particular one has

φ_{1}(X) = cardinality of X

if 1 = {1_{G}} denotes the trivial subgroup of G. Essential properties of the canonical homomorphisms

• For U, V ≤ G one has φ_{U} = φ_{V} if and only if U and V are conjugate

in G (U∼^{G}V).

• For x, x′ ∈ Ω(G) one has φ_{U}(x) = φ_{U}(x′) for all U ≤ G if and only if x = x′.

The canonical injection of the Burnside ring into the ghost ring Consider for each x ∈ Ω(G) to the map

U ⟼ φ_{U}(x)

from the set Sub(G) of all subgroups of G into Z. Then

x ↦ (U ⟼ φ_{U}(x))

provides a canonical homomorphism φ : Ω(G) ⟶ Ω̃(G) from the Burnside ring into the ghost ring

Ω̃(G) := Z^{Sub(G)/ ∼}

of G, consisting of all maps from Sub(G) into Z which are constant on each conjugacy class of subgroups. This homomorphism is injective and allows to interpret the Burnside ring Ω(G) as a subring of the ghost ring Ω̃(G). The canonical basis of the Burnside ring For every subgroup U of the group G one has the G–set G/U := {gU | g ∈ G} of left cosets of G modulo U. It is transitive and every transitive G–set is isomorphic to one of this form. Since every G–set decomposes uniquely into a disjoint union of transitive G–sets, the (isomorphism classes of the) coset spaces G/U form a Z–basis of Ω(G) and every element x ∈ Ω(G) can be expressed as a linear combination in the form

x = \begin{matrix}∑ & ′ \\ U≤G \end{matrix} μ_{U}(x) · G/U

with uniquely determined integral coefficients μ_{U}(x) ∈ Z, satisfying μ_{U}(x) = μ_{V} (x) for U∼^{G}V. The prime attached to the summation symbol ^{∑ ′} indicates that the sum extends only over a system of representatives of the conjugacy classes of subgroups of G. Calculating Burnside numbers For every element x of the G–set X denote by G_{x} := {g ∈ G | gx = x} the isotropy group of x.

• If f : X ⟶ Y is a G–map mapping the element x ∈ X to the element y ∈ Y then G_{x} ≤ G_{y}.

• Vice versa:

If X is a transive G–set and if G_{x} ≤ G_{y} then there exists a unique G–map mapping x to y.

• This provides a canonical bijection

Hom_{G}(G/V, X) ≅ X^{V} f ↦ f(V)

which specializes for Y = G/U to

(G/U)^{V} ≅ Hom_{G}(G/V, G/U)

• Hence:

φ_{V} (G/U) ≠ 0 if and only if there exists some element g ∈ G with V ⊆ gUg^{−1}, i. e. if V is sub–conjugate to U (V ∼^{<}_{G}U).

Note that Aut_{G}(G/U) operates freely on G/U and that Aut_{G}(G/U) ≅ N_{G}(U)/U. Therefore

φ_{V} (G/U) = #{gU ∈ G/U | V gU = gU}
= (N_{G}(U) : U) · #{U′ ≤ G | V ≤ U′∼^{G}U}.

Hence for a given element x ∈ Ω(G) a subgroup U of G

• is a maximal subgroup with μ_{U}(x) ≠ 0

if and only if

• it is a maximal subgroup with φ_{U}(x) ≠ 0

and for such a maximal subgroup one has

φ_{U}(x) = μ_{U}(x) · φ_{U}(G/U) = μ_{U}(x) · (N_{G}(U) : U).

An application for p–groups

The unique representation

x = \begin{matrix}∑ & ′ \\ U≤G \end{matrix} μ_{U}(x) · G/U

existing for every x ∈ Ω(G) implies in case that G is a p–group:

φ_{1}(x) = \begin{matrix}∑ & ′ \\ U≤G \end{matrix} μ_{U}(x) · (G : U)
≡ μ_{G}(x) = φ_{G}(x)  (mod p).

Corollary: If V is a p–subgroup of an arbitrary finite group G and if U is a subgroup of G with an index (G : U) which is prime to p, then

φ_{V} (G/U) ≡ φ_{1}(G/U) = (G : U) ≢ 0 (mod p)

and therefore V is sub–conjugate to U (V ∼^{<}_{G}U).

Corollary: If Sylow p–subgroups exist in G, they all must be conjugate in G and every other p–group must be sub–conjugate in G to each of them.

Proof of the Theorem

Let β denote the map from Sub(G) into Sub(C) which associates to every subgroup U of G the unique subgroup C_{|U|} of C which has the same number of elements as U. Then clearly, β induces a ring homomorphism

γ = γ(G) : Ω̃(C) ⟶ Ω̃(G) s ⟼ s ∘ β

from the ghost ring of C into the ghost ring of G such that

φ_{U}(α(x)) = φ_{C_{|U|}}(x).

Hence the Theorem just claims that γ maps the subring Ω(C) of Ω̃(C) into the subring Ω(G) of Ω̃(G) and that α is precisely the restriction of γ onto Ω(C).

To prove that γ(Ω(C)) is already contained in Ω(G) recall that for every

finite G–set X and every natural number q the set \binom{X}{q} of all subsets Y of

X of cardinality q is also a finite G–set relative to the G–action

G × \binom{X}{q} ⟶ \binom{X}{q} : (g, Y) ⟼ g · Y := \{g · y | y ∈ Y \}.

Using these G–sets for the regular G–set X := G/1 the Theorem follows immediately from the following two observations:

Lemma 1:

γ maps \binom{C/1}{q} ∈ Ω(C) ⊆ Ω̃(C) onto \binom{G/1}{q} ∈ Ω(G) ⊆ Ω̃(G).

Lemma 2: If J_{n} := {d ∈ N | d divides n} denotes the set of divisors of n then the family

\binom{C/1}{d} (d ∈ J_{n}) of C–sets forms a Z–basis of Ω(C).

Lemma 1 in turn is an immediate consequence of the well–known fact that

the Burnside number φ_{U}(\binom{G/1}{q}), that is, the number of U–invariant subsets

of cardinality q in G/1, depends only on q and the orders of G and of U. This fact is expressed more explicitly in the following Lemma 1′: For every finite group G, every subgroup U of G, and every q ∈ N one has

φ_{U}(\binom{G/1}{q}) = \{ \begin{matrix}0 & if |U| does not divide q, \\ \binom{(G : U)}{q/|U|} & otherwise, \end{matrix}

In particular, if |U| = q, then φ_{U}(\binom{G/1}{q}) = (G : U) and therefore

μ_{U}(\binom{G/1}{q}) = \frac{(G : U)}{(N_{G}(U) : U)} = (G : N_{G}(U)).

Proof of Lemma 1′:

• Y ∈ \binom{G/1}{q} is U–invariant if and only if Y is the union of right cosets Ug ⊆ G of U in G.

• Hence such a subset Y exists only if |U| divides q.

• In this case the set \binom{G/1}{q}^{U} of U–invariant subsets Y in \binom{G/1}{q} cor-

responds in a one–to–one fashion to the set \binom{U\G}{q/|U|} of subsets of U\G := {Ug | g ∈ G} of cardinality q/|U|.

• So its cardinality is of course \binom{(G : U)}{q/|U|}, as stated.

Proof of Lemma 2: For every d, d′ ∈ J_{n} we have integers

μ_{d,d′} = μ_{C_{d′}}(\binom{C/1}{d}) ∈ Z

such that

\binom{C/1}{d} = ∑_{d′∈J_{n}} μ_{d,d′} · C/C_{d′}

and we have to show that the determinant of the matrix

M := (μ_{d,d′})_{d, d′ ∈ J_{n}}

is a unit in Z.

In view of Lemma 1 we have

• φ_{C_{d′}}(\binom{C/1}{d}) = 0 unless d′ divides d.

• Hence, we have also μ_{d,d′} = 0 unless d′ divides d.

• Therefore M is a triangular Matrix (relative to the obvious ordering of J_{n} according to which d comes before d′ if d is smaller than d′).

• In addition, we have

μ_{d,d} = μ_{C_{d}}(\binom{C/1}{d}) = (C : N_{C}(C_{d})) = 1,

so the main diagonal of M consists of one’s, only.

Hence the determinant of the matrix M is indeed equal to 1.

Remark: Rather than using exterior powers of G–sets, that is, the G–sets of the form

\binom{G/1}{q}, introduced by H. Wielandt [Wi 59] ] in this context, we could as

well have used the symmetric powers, that is, the G–sets of the form

S^{q}(X) := {f : X ⟶ N_{0} | ∑_{x∈X} f(x) = q},

used by B. Wagner [Wa 80]. As before, the value of φ_{U}(S^{q}(G/1)) depends only on q, |G|, and |U| and vanishes unless |U| divides q, and the family S^{d}(C/1) (d ∈ J_{n}) of C–sets forms a Z–basis of Ω(G).

Corollary 1: For every divisor d of |G| there exists an element x_{d} ∈ Ω(G) satisfying

φ_{U}(x_{d}) = \{ \begin{matrix}d & if d divides (G : U), \\ 0 & otherwise, \end{matrix}

In particular, μ_{U}(x_{d}) = 0 unless d divides (G : U) and

μ_{U}(x_{d}) = (G : N_{G}(U)) = #{gUg^{−1} | g ∈ G}

if (G : U) = d.

Proof of Corollary 1: Put x_{d} := α(C/C_{|G|/d}). Then one has

• φ_{U}(x_{d}) = φ_{C_{|U|}}(C/C_{|G|/d}) = (C : C_{|G|/d}) = d if C_{|U|} ⊂ C_{|G|/d} that is, if the index d of C_{|G|/d} in C divides the index (C : C_{|U|}) = (G : U) of C_{|U|} in C.

• φ_{U}(x_{d}) = 0 otherwise and therefore also μ_{U}(x_{d}) = 0 if d does not divide (G : U).

• If (G : U) = d, then U is a maximal subgroup of G with μ_{U}(x_{d}) ≠ 0 and therefore

μ_{U}(x_{d}) = \frac{φ_{U}(x_{d})}{(N_{G}(U) : U)} = \frac{d}{(N_{G}(U) : U)}
= \frac{(G : U)}{(N_{G}(U) : U)} = (G : N_{G}(U))

equals the number of subgroups in G which are conjugate to U in G.

To derive the next three corollaries we follow essentially the ideas of B. Wagner [Wa 80].

Corollary 2 (Sylow): Every divisor d of |G| is the greatest common divisor of all indices (G : U) of subgroups U in G which are divisible by d, that is, we have

d = g.c.d. ( (G : U) | d divides (G : U)) .

In particular (or, as well, equivalently),

• if |G| = d · p^{α} for some prime p, then there exist subgroups U of G of index d and hence of order p^{α}.

In case |G|/d is a power of a prime p we can exploit this argument even further to derive:

Corollary 3 (Sylow, Frobenius): If a power p^{α} of a prime p divides the order |G| of a finite group G, then the number of subgroups V of order p^{α} is congruent to 1 modulo p.

Proof of Corollary 2: Write x_{d} ∈ Ω(G) in the form

x_{d} = \begin{matrix}∑ & ′ \\ U≤G \end{matrix} μ_{U}(x_{d}) · G/U = \begin{matrix}∑ & ′ \\ U≤G, d|(G:U) \end{matrix} μ_{U}(x_{d}) · G/U

and apply φ_{1} to derive

d = \begin{matrix}∑ & ′ \\ U≤G,d|(G:U) \end{matrix} μ_{U}(x_{d}) · (G : U)

and consequently

d ∈ \begin{matrix}∑ & ′ \\ U≤G,d|(G:U) \end{matrix} Z · (G : U).

Proof of Corollary 3: Put d := |G|/p^{α} and divide the above equation

d = \begin{matrix}∑ & ′ \\ U≤G, d|(G:U) \end{matrix} μ_{U}(x_{d}) · (G : U)
= \begin{matrix}∑ & ′ \\ U≤G, |U|∈J_{p^{α}} \end{matrix} μ_{U}(x_{d}) · (G : U)

by d to derive

1 = \begin{matrix}∑ & ′ \\ U≤G,d|(G:U) \end{matrix} μ_{U}(x_{d}) · \frac{(G : U)}{d}

= \begin{matrix}∑ & ′ \\ U≤G,|U|∈J_{p^{α}} \end{matrix} μ_{U}(x_{d}) · \frac{p^{α}}{|U|}
≡ \begin{matrix}∑ & ′ \\ U≤G,|U|=p^{α} \end{matrix} μ_{U}(x_{d})
= \begin{matrix}∑ & ′ \\ U≤G,|U|=p^{α} \end{matrix}(G : N_{G}(U))
= \begin{matrix}∑ & ′ \\ U≤G,|U|=p^{α} \end{matrix} #\{gUg^{−1} | g ∈ G\}
= #{V ≤ G | |V | = p^{α}}  (mod p).

Cauchy–Frobenius–Burnside congruence relations To derive the next corollary let us recall that for every x ∈ Ω(G) one has the so called

Cauchy–Frobenius–Burnside congruence relation

∑_{g∈G} φ_{⟨g⟩}(x) ≡ 0 (mod |G|).

By additivity it is enough to verify this just for x = G/U (U ≤ G) in which case a standard computation yields

∑_{g∈G} φ_{⟨g⟩}(G/U) = ∑_{g∈G} #{hU ∈ G/U | ghU = hU}
= ∑_{hU∈G/U} #{g ∈ G | ghU = hU}
= ∑_{hU∈G/U} |hUh^{−1}|
= (G : U) · |U| = |G|
≡ 0 (mod |G|).

with the Cauchy–Frobenius–Burnside relation Corollary 1 yields Corollary 4 (Frobenius): Every divisor m of the order |G| of a finite group G divides also the number

#{g ∈ G | g^{m} = 1}

of elements g in G, whose order divides m. Proof: Apply the Cauchy–Frobenius–Burnside congruence relation to x_{d} for d := |G|/m to derive that |G| = d · m divides

∑_{g∈G} φ_{⟨g⟩}(x_{d}) = ∑_{g∈G,d|(G:⟨g⟩)} d
= ∑_{g∈G,|g|∈J_{m}} d
= d · #{g ∈ G | g^{m} = 1}

and hence, dividing by d, that m divides #{g ∈ G | g^{m} = 1}. For more details, further applications and detailed references see [DSY 90].

References

[DSY 90] Dress (A. W. M.), Siebeneicher (Ch.), Yoshida (T.) — An Application of Burnside Rings in Elementary Finite Group Theory — Preprint 90–033 des SFB 343 Diskrete Strukturen in der Mathematik, Bielefeld 1990, to appear in Adv. in Math.

[Fr 95] Frobenius (G.) — Verallgemeinerung des Sylowschen Satzes — Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1895) 981–993, in Gesammelte Abhandlungen Bd. II, Springer–Verlag, Berlin–New York, 1968, 664–676.

[Fr 03] Frobenius (G.) — Über einen Fundamentalsatz der Gruppentheorie — Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1903) 987–991, in Gesammelte Abhandlungen Bd. III, Springer–Verlag, Berlin–New York, 1968, 330–334.

[Fr 07] Frobenius (G.) — Über einen Fundamentalsatz der Gruppentheorie — Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1907) 428–437, in Gesammelte Abhandlungen Bd. III, Springer–Verlag, Berlin–New York, 1968, 428–437.

[Hu 67] Huppert (B.) — Endliche Gruppen I — Springer Verlag, Berlin– Heidelberg–New York, 1967, p. 34.

[Sy 72] Sylow (L.) — Théorèmes sur les groupes de substitutions — Math. Ann., 5 (1872), 584–594.

[Wa 80] Wagner (B.) — A permutation representation theoretical version of a theorem of Frobenius — Bayreuther Mathematische Schriften 6 (1980), 23–32.

[Wi 59] Wielandt (H.) — Ein Beweis für die Existenz der Sylowgruppen — Arch. der Math., 10 (1959), 401 – 402.