Journal of Lie Theory Volume 12 (2002) 305–308 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

The Abelian Subgroup Conjecture: A Counter Example

Wolfgang Herfort

Communicated by K. H. Hofmann

Abstract. If an abelian subgroup A of a locally compact group G has the same weight as G, it is termed large (see [3]). It has been conjectured that every compact group has a large abelian subgroup. In this note we show that no free pro-p group F (X) on set X of cardinality greater than ℵ_{0} contains a large abelian subgroup.

1. The Background

The weight w(G) of a locally compact topological group is defined as the minimal cardinality of a base of open sets. K. H. Hofmann and S. A. Morris in [3] proposed the following:

Conjecture 1.1. (The Abelian Subgroup Conjecture) Every infinite compact group G has an abelian subgroup A whose weight equals that of G.

For brevity, we adopt the convention in [3], to say G is an LAS-group, if it satisfies the conjecture. As an important consequence of their investigations the authors prove the following extension Theorem (Theorem D):

Theorem 1.2. Let G be an infinite compact group and N a closed subgroup such that G/N and N are LAS-groups. Then G is an LAS-group.

In particular, when N = G_{0} is the connected component of G they show in Corollary E:

Corollary 1.3. Let G be a compact group and assume that G/G_{0} is an LASgroup. Then G is an LAS-group.

Therefore, if the Abelian Subgroup Conjecture is false, it should fail for some profinite group. According to [2], a compact group G is strictly reductive, if it is isomorphic to the direct product of simple compact groups (a compact group is called simple, if it does not contain nontrivial closed normal subgroups). Then in [3] the same authors show in Corollary F:

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Corollary 1.4. Every strictly reductive compact group is an LAS-group.

The most important step in further reducing the Abelian Subgroup Conjecture is Theorem 3.4, the Countable Layer Theorem, in [2], which, together with Theorem 4.4, the Topological Decomposition Theorem, implies the following theorem:

Theorem 1.5. Every profinite group G has a series G = G^{0} > G^{1} > … of characteristic closed subgroups with each G^{i}/G^{i−1} strictly reductive and ⋂_{i∈ℕ} G^{i} = {1}. Moreover w(G) = sup_{i∈ℕ}{w(G^{i}/G^{i+1}).

Proof. The last statement is a consequence of Theorem 4.14 in [2].

2. The Counterexample

Profinite groups admit, for every prime p, p-Sylow subgroups, any two of them being conjugate, and every pro-p subgroup of G being contained in a suitable p-Sylow subgroup. The cofinality of a cardinal ℵ is the smallest cardinal ℵ′ such that the set of predecessors of ℵ contains a subset S of cardinality ℵ′ satisfying ℵ = sup S.

Lemma 2.1. Let G = ∏_{j∈J} S_{j} with simple compact groups S_{j}.

(i) If w(G) has uncountable cofinality, then G has either a torus subgroup T or a p-Sylow subgroup P for a some prime p such that w(T) = w(G) or w(P) = w(G), respectively.

(ii) If the cardinal ℵ has countably infinite cofinality, then a family of groups S_{j} of prime order can be selected so that G has weight ℵ and all its Sylow-subgroups have a weight that is properly smaller than ℵ.

Proof. In both cases w(G) is infinite. We assume w(G) ≥ ℵ_{0}. The identity component G_{0} is a product of simple Lie groups. It has a maximal torus T such that w(T) = w(G_{0}). Let S be a system of representatives of the finite simple groups; note that S is countable. Then G = G_{0} × ∏_{S∈S} S^{J(S)} for a countable family of cardinals J(S). Then w(G) = max {ℵ_{0}, w(G_{0}), sup{card J(S) : S ∈ S}}

Assume now that w(G) has uncountable cofinality. Then, since S is countable, either w(G) = w(G_{0}) = w(T) or there is an S ∈ S such that w(G) = card J(S). In the second case let p be any prime such that S has a nontrivial p-Sylow subgroup. Then S^{J(S)} contains a subgroup isomorphic to ℤ(p)^{w(G)} and thus G has a p-Sylow subgroup of weight w(G).

Assume next that ℵ is an infinite cardinal of countable cofinality. Let {ℵ_{n} : n ∈ ℕ} be a sequence of infinite cardinals ℵ_{n} < ℵ such that ℵ = sup_{n∈ℕ} ℵ_{n}. Let {p_{n} : n ∈ ℕ} = {2, 3, 5, 7, … } be the sequence of primes in ascending order. Set G = ∏_{n∈ℕ} ℤ(p_{n})^{ℵ_{n}}. The weight of the p_{n}-Sylow subgroup ℤ(p_{n})^{ℵ_{n}} of G is ℵ_{n} < ℵ = sup_{m∈ℕ} ℵ_{m} = w(G).

Lemma 2.2. For any profinite group G whose weight w(G) has uncountable cofinality there exists a p-Sylow subgroup G_{p} with w(G_{p}) = w(G).

Proof. The Countable Layer Theorem 1.5 yields w(G) = sup_{i∈N} w(G_{i}/G_{i+1}). Since w(G) has infinite cofinality, there is a natural number i such that w(G) = w(G_{i}/G_{i+1}). The hypotheses of 2.1 apply to G_{i}/G_{i+1} and secure the existence of a prime p such that the p-Sylow subgroup P_{i} of G_{i}/G_{i+1} satisfies w(P_{i}) = w(G). Let P be the p-Sylow subgroup of G. The image P G_{i+1}/G_{i+1} of the p-Sylow subgroup P of G in the quotient G/G_{i+1} is a p-Sylow subgroups. Thus it contains a conjugate of the p-group P_{i}. Therefore w(P) ≥ w(P G_{i+1}/G_{i+1}) ≥ w(P_{i}) = w(G). Trivially, w(P) ≤ w(G). Hence w(P) = w(G). This proves the lemma.

Thus, if there is any profinite counterexample to Conjecture 1.1, there should be one which is a pro-p group and we turn to describing one of them.

For this purpose recall the concept of a free pro-p group: Let X be any set, then there is a pro-p group F_{p}(X), unique up to isomorphism, containing X in such a way that every open normal subgroup contains all but a finite number of elements of X, and having the following universal property: Assume that f: X → G is any function into a profinite group such that for every open normal subgroup N of G, the identity neighborhood f^{−1}(N) contains all but a finite number of the elements of X; then there is a unique morphism of topological groups F_{p}(X) → G extending f. The details can be found e.g. in Chapter 3 of [4] and in Chapter 5 [5].

Example 2.3. Fix any prime p and an infinite set X. Then F_{p}(X) is a compact group of weight card(X) such that every closed abelian subgroup is isomorphic to the additive group of p-adic integers ℤ_{p}, whose weight is ℵ_{0}.

In particular, whenever card(X) > ℵ_{0}, F_{p}(X) is a counterexample to Conjecture 1.1.

Proof. Since |X| is infinite, [4], Proposition 2.6.1 (b) together with Proposition 2.6.2, or [5], p. 84, Lemma 5.5.1, yield w(F_{p}(X)) = |X|. Let A be any maximal abelian subgroup of F_{p}(X). Then, by Corollary 7.7.5 in [4], or by [5], p. 83, Theorem 5.4.6, or by [1], the subgroup A is again a free pro-p group; since A is abelian and nonsingleton, A ≅ ℤ_{p}.

Aacknowledgement: The author would like to thank K. H. Hofmann for lively email discussions, contributing Lemma 2.1 and adjusting the original proof of Lemma 2.2.

References

[1] Cossey, J., G. H. Kegel, and L. G. Kovács, Maximal Frattini extension, Arch. Math. 35 (1980), 210–217.

[2] Hofmann, K. H., and S. A. Morris, A structure theorem on compact groups, Math. Proc. Cambridge Philos. Soc. 130 (2001), 409–426.

[3] —, Compact groups with large abelian subgroups, Math. Proc. Cambridge Philos. Soc. 133 (2002), to appear.

[4] Ribes, L., and P. A. Zalesskii, Profinite Groups, Springer-Verlag, Berlin etc., 2000, xiv+435 pp.

[5]  Wilson, J. S., “Profinite Groups,” Oxford Univ. Press, 1998, xi+284pp.

Wolfgang Herfort University of Technology Vienna, Austria herfort@tuwien.ac.at

Received August 30, 2001 and in final form October 10, 2001