Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 1, 147-148.

Still another Proof of the Existence of Sylow–p–Subgroups

Andreas W.M. Dress

Forschungsschwerpunkt Mathematisierung - Strukturbildungsprozesse Fakultät für Mathematik, Universität Bielefeld Postfach 10 01 31, D - 33501 Bielefeld

The classical inductive proof of the existence of Sylow–p–subgroups became obsolete when H. Wielandt published his ingenious non–inductive proof in 1958 (cf. [4], see also [3] for a more structural interpretation of Wielandt’s approach). In this note we want to point out still another simple, though inductive proof for the existence of Sylow–p–subgroups which is based on the essentially trivial observation that for every non–trivial subgroup 1 ≠ U ≤ G of a finite group G the set X(U) := {f : G/U → G | f(gU) ϵ gU for all g ∈ G} of all systems of coset representatives of U in G is a fixed point free G–set of cardinality |X(U)| = |U|^{(G:U)} and hence, by orbit decomposition, there exist proper subgroups V_{i} < G of G (i = 1,…, n_{U}) with

|U|^{(G:U)} = ∑_{i}(G : V_{i}).

Indeed, assume that there exists some finite group G and some prime p for which no subgroup U ≤ G of order |G|_{p} := max (p^{α} | p^{α} divides |G|) exists and assume that among all such pairs the pair (G, p) is chosen so that G has minimal order.

Then this order can’t be a prime power, in particular it is not a prime and so G has at least one proper, non–trivial subgroup 1 ≠ U < G. Since |U| < |G|, there exists a subgroup V ≤ U with |V | = |U|_{p}, hence |U|_{p} ≠ |G|_{p}, that is p divides (G : U) for all proper subgroups U. Therefore, by the above observation, p divides |U| for every non– trivial subgroup 1 < U < G, so a proper subgroup U of G cannot contain a non–trivial Sylow–q–subgroup for every prime q ≠ p and hence, in view of the minimality of |G|, we must have |U|_{q} = 1 for every prime q ≠ p, that is, every proper subgroup of G must be a p–group. But this implies that |G|_{p′} := |G|/|G|_{p} divides (G : V) for every proper subgroup V < G of G and so, using the above observation once more, it must divide the p–power |U|^{(G:U)} for every proper, non–trivial subgroup 1 ≠ U < G, a final contradiction.

The argument, we have actually used in this proof, can be put in the following more abstract form:

Lemma: Assume that for some n ∈ N a subset P of the set

P (n) := {d ∈ N | d divides n}

0138-4821/94 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1993 Heldermann Verlag, Berlin

of divisors of n is given such that the following holds:

• 1, n ∈ P,

• for every d ∈ P which is not a prime there exists a non–empty subset

P_{d} ⊆ P ∩ P (d) \ {1, d}

such that for every t ∈ P_{d} the power t^{d/t} can be written in the form

(∗)  t^{d/t} = ∑_{i} d/t_{i}

for appropriate t_{1}, t_{2},… ∈ P_{d}. Then for every d ∈ P and every prime p the p–part d_{p} of d is also in P.

Proof: Otherwise, there exists a minimal d ∈ P for which there exists some prime p with d_{p} ∉ P, in particular 1 ≠ d_{p} ≠ d. It follows that P_{d} is non–empty and that t_{p} ∈ P for every t ∈ P_{d}. Hence, p must divide the quotient d/t for every t ∈ P_{d} and therefore, in view of (∗), it must divide every t ∈ P_{d}. So, using the minimality of d once more, this time with respect to primes q ≠ p, every t ∈ P_{d} must be a p–power, so 1 ≠ d_{p′} := d/d_{p} must divide every quotient d/t (t ∈ P_{d}) and therefore, using (∗) once more, it must divide the numbers t^{d/t} which we have just recognized as p–powers, — the final contradiction.

Since the above observation ensures that for every finite group G of order n the set P_{G} := {|U| | U ≤ G} satisfies the condition of the Lemma (for d ∈ P_{G} define P_{d} := ⋃_{U≤G,|U|=d} P_{U} \ {1, d}), the existence of Sylow–p–subgroups can now be derived as its corollary.

From a (permutation–)group theoretic point of view it might be more interesting to study the orbit structure of X(U) more explicitly, noting that it plays a crucial rôle in multiplicative transfer (cf. for example [1, Theorem 5, p. 222]) and related considerations (cf. [2, §5]).

References

[1] A.W.M. Dress: Contributions to the theory of induced representations. Proceedings of the Conf. on Alg. K–Theory, Seattle, 1972, Springer Lecture Notes 342, 183 – 240 (1973).

[2] A.W.M. Dress and Chr. Siebeneicher: The Burnside ring of profinite groups and the Witt vector construction. Advances in Mathematics 70, 87 – 132 (1988).

[3] A.W.M. Dress, Chr. Siebeneicher and T. Yoshida: An Application of Burnside Rings in Elementary Finite Group Theory. Advances in Mathematics 91, 27 – 44 (1992).

[4] H. Wielandt: Ein Beweis für die Existenz der Sylowgruppen. Archiv der Mathematik 10, 401 – 402 (1959).

Received 20.10.93