Turk J Math () , 1 – 8. \begin{matrix}c \\ ◯ \end{matrix} TÜBİTAK

A non-linear locally finite simple group with a p-group as centralizer

U. Meierfrankenfeld Michigan State University meier@math.msu.edu

Abstract

We show that there exists a non-linear, locally finite, simple group such that the centralizer of every non-trivial element is (locally solvable)-by-finite.

In [1, Problem. 3.8] Brian Hartley asked the following question:

Does there exists a non-linear infinite simple locally finite group in which the centralizer of every non-trival element is almost soluble, that is, has a soluble subgroup of finite index?

In this note we will give a partial answer to Brian’s question: We will show that the answer is affirmative if solvable is replaced by “locally solvable”. More precisely we prove:

Theorem A (a) There exists a non-linear, locally finite, simple group such that the centralizer of every non-trivial element is (locally solvable)-by-finite.

(b) Let p be a prime. Then there exists a non-linear, locally finite, simple group with an element whose centralizer is a p-group.

This theorem and its proof first appeared in my lecture notes on locally finite simple groups [2].

If W and I are sets, then W^{I} denotes the set of functions from I to W. If X is group, Y a subgroup of X and W a Y -set we define

W⇑_{Y}^{X}= {f : X → W | f(xy) = f(x)^{y} for all x ∈ X, y ∈ Y }.

Note that, if we view X as a Y -set by right multiplication, then W ⇑_{Y}^{X} just consists of the Y -equivariant maps from X to W.

If W is a Y -module and X/Y is finite, then the following lemma shows that

W⇑_{Y}^{X} is the induced module for X. And if W is a group with Y acting trivially on

W, then W⇑_{Y}^{X} is the base group of the wreath product W ≀_{X/Y} X.

Lemma 1 Let X be a group, Y a subgroup of X and W a Y -set. Put V = W⇑_{Y}^{X}. Then

(a) X acts on V by f^{h}(x) = f(hx) for all x, h ∈ X.

(b) Let I be a left transversal to Y in X. Then the restriction map

ρ_{I} : V → W^{I}, f → f |_{I}

is a bijection. In particular, V and W^{X/Y} are isomorphic as sets.

(c) Define

π : V → W, π(f) = f(1).

Then π is an onto Y -equivariant map.

(d) Suppose that t is a fixed-point for Y on W. Let w ∈ W and define

κ_{t}(w) : X → W, x → \{ \begin{matrix}w^{x} & if x ∈ Y \\ t & if x ∉ Y \end{matrix} .

Then

(a) κ_{t}(w) ∈ V and κ_{t} : W → V, w → κ_{t}(w) is a 1-1 Y -equivariant map.

(b) π(κ_{t}(w)) = w.

(c) π(κ_{t}(w)^{x})) = t for all x ∈ X \ Y.

(e) Suppose in addition that W is a Y -group, that is, W is a group and for each y ∈ Y the map W → W, w → w^{y} is a homomorphism of groups. Then the maps ρ_{I}, π and κ_{1} all are homomorphism of groups.

Proof. (a): We need to verify that f^{h} ∈ V and f^{hl} = (f^{h})^{l} for all f ∈ V, h, l ∈ X. Let x ∈ X and y ∈ Y. Since f^{h}(xy) = f(h(xy)) = f((hx)y) = f(hx)^{y} = (f^{h}(x))^{y}, f^{h} ∈ V. Also f^{hl}(x) = f((hl)x) = f(h(lx)) = f^{h}(lx) = (f^{h})^{l}(x) and so (a) holds.

(b): Let x ∈ X. Then x = iy for some unique i ∈ I, y ∈ Y.

Let f ∈ V. Then f(x) = f(iy) = f(i)^{y} and so f is uniquely determined by f_{I}. Thus ρ_{I} is 1-1.

Let g ∈ W^{I}. Define f : X → W by f(x) = f(i)^{y}. It is easy to verify that f ∈ V and f_{I} = g. Hence ρ_{I} is onto.

(c): Let f ∈ V, y ∈ Y. Then π(f^{y}) = f^{y}(1) = f(y·1) = f(1·y) = f(1)^{y} = π(f)^{y}. So π is Y -equivariant. Choose a left transversal containing 1. Then (b) implies that π is onto.

(d): Let w ∈ W, y ∈ Y and x ∈ X. Then x ∈ Y if and only if xy ∈ Y. Also by assumption t = t^{y} and so

κ_{t}(w)(xy) = \{ \begin{matrix}w^{xy} & if xy ∈ Y \\ t & if xy ∉ Y \end{matrix} = \{ \begin{matrix}(w^{x})^{y} & if x ∈ Y \\ t^{y} & if x ∉ Y \end{matrix} = (κ_{t}(w)(x))^{y}.

Thus κ_{t}(w) ∈ V.

Also yx ∈ Y if and only if x ∈ Y. So

κ_{t}(w^{y})(x) = \{ \begin{matrix}(w^{y})^{x} & if x ∈ Y \\ t & if x ∉ Y \end{matrix} = \{ \begin{matrix}w^{yx} & if yx ∈ Y \\ t & if yx ∉ Y \end{matrix} = κ_{t}(w)(yx) = κ_{t}(w)^{y}(x).

Thus κ_{t}(w^{y}) = κ_{t}(w)^{y} and (d:a) holds.

Since 1 ∈ Y, π(κ_{t}(w)) = κ_{t}(w)(1) = w^{1} = w and so (d:b) holds.

Let x ∈ X \ Y. Then π(κ_{t}(w)^{x}) = κ_{t}(w)^{x}(1) = κ_{t}(w)(x · 1) = κ_{t}(w)(x) = t. So also (d:c) holds.

It remains to prove (e). So suppose that W is a Y -group. Clearly W^{X} is a group

via (fg)(x) = f(x)g(x). Moreover, for f, g ∈ W⇑_{Y}^{X}, x ∈ X and y ∈ Y we have

(fg)(xy) = f(xy)g(xy) = f(x)^{y}g(x)^{y} = (f(x)g(x))^{y} = (fg)(x)^{y}.

So fg ∈ W ⇑_{Y}^{X}. Similarly f^{−1} ∈ W ⇑_{Y}^{X} and clearly 1 ∈ W ⇑_{Y}^{X}. Hence W ⇑_{Y}^{X} is a

subgroup of W^{X}.

For any J ⊆ X, the restriction map W^{X} → W^{J}, f → f |_{J} is a homomorphism. Thus ρ_{I} and π are homomorphism. W → W, w → w^{y} and W → W, w → 1 are homomorphisms and so also κ_{1} is a homomorphism. □

Statement (c:a) in the following lemma is crucial for this paper. It allows us to enlarge a group Y to a group H while controlling some of the centralizers.

Lemma 2 Let X, Y, W, G be groups with Y ≤ X, G ≤ Y, W ⊴ Y, Y = W G and W ∩ G = 1. Then there exists a semidirect product H = V ⋊ X and an embedding β : Y → H such that

(a) V ≅ W^{Y/X} as groups.

(b) V β(y) = V y for all y ∈ Y.

(c) Let y ∈ Y, w ∈ W and g ∈ G with y = wg and y ∉ g^{W}. Then

(a) V C_{H}(β(y)) = V C_{Y} (y).

(b) β(y) ∉ y^{V}.

Proof. Let y ∈ Y and y = wg with w ∈ W, g ∈ G. Let ρ be the projection of Y onto G, that is ρ(wg) = g. Note that W is a Y -group via w → w^{ρ(y)}. We denote this Y -group by W_{ρ}. Put V = W_{ρ} ⇑_{Y}^{X}. Then by Lemma 1 X acts on V and we can form the semidirect product, H = V ⋊ X = {(v, x)|v ∈ V, x ∈ X}. We view V and X as subgroups of H. So H = V X and V ∩ X = 1. Let π : V → W and κ = κ_{1} : W → V be as in Lemma 1. Let v ∈ V.

1^{∘} π(v^{y}) = π(v)^{ρ(y)} = π(v)^{g} and π(v^{y^{−1}}) = π(v)^{g^{−1}}.

The first statement follows from 1(c) and the definition of action of Y on W_{ρ}. Since ρ(y^{−1}) = ρ(y)^{−1} = g^{−1}, the second statement follows from the first. 2^{∘} Define

β : Y → H, y → (κ(w), y).

Then β is a monomorphism and V β(y) = V y.

Clearly β is 1-1 and V β(y) = V y. For i = 1, 2 let y_{i} ∈ Y and y_{i} = w_{i}g_{i} with w_{i} ∈ W, g_{i} ∈ G. Then

y_{1}y_{2} = w_{1}g_{1}w_{2}g_{2} = w_{1}w_{2}^{g_{1}^{−1}}g_{1}g_{2},

and so

β(y_{1}y_{2}) = (κ(w_{1}w_{2}^{g_{1}^{−1}}), y_{1}y_{2}).

On the other hand,

β(y_{1})β(y_{2}) = (κ(w_{1}), y_{1})(κ(w_{2}), y_{2}) = (κ(w_{1})κ(w_{2})^{y_{1}^{−1}}, y_{1}y_{2}).

As κ is a Y -equivariant homomorphism,

β(y_{1})β(y_{2}) = (κ(w_{1}w_{2}^{g_{1}^{−1}}), y_{1}y_{2})

and so β is a homomorphism.

3^{∘} Let (v, x) ∈ C_{H}(β(y)). Then κ(w)v^{y^{−1}} = vκ(w)^{x^{−1}} and xy = yx. We compute

β(y)(v, x) = (κ(w), y)(v, x) = (κ(w)v^{y^{−1}}, yx)

and

(v, x)β(y) = (v, x)(κ(w), y) = (vκ(w)^{x^{−1}}, xy)

Thus (3^{∘}) holds.

4^{∘} Suppose that V C_{H}(β(y)) ≠ V C_{Y} (y). Then y ∈ g^{W}.

Since β(C_{Y} (y)) ≤ C_{H}(β(y)) we have V C_{Y} (y) =^{(2^{∘})} V β(C_{Y} (y)) ≤ V C_{H}(β(y)) and so there exists (v, x) ∈ C_{H}(β(y)) with x ∉ Y.

By Lemma 1(d:b), π(κ(w)) = w. By (1^{∘}), π(v^{y^{−1}}) = π(v)^{k^{−1}}. Also since x ∉ Y and κ = κ_{1}, Lemma 1(d:c) implies π(κ(w)^{x^{−1}}) = 1. So applying π to both sides of the first equation in (3^{∘}) we obtain

wπ(v)^{g^{−1}} = π(v).

Put r = π(v). Then wgrg^{−1} = r, wg = rgr^{−1} and y = wg = g^{r^{−1}} ∈ g^{W}.

5^{∘} If β(y) ∈ y^{V}, then y ∈ g^{W}.

Suppose that β(y) = (1, y)^{(v,1)} for some v ∈ V. Then

(κ(w), y) = (v^{−1}, 1)(1, y)(v, 1) = (v^{−1}, y)(v, 1) = (v^{−1}v^{y^{−1}}, y)

and so κ(w) = v^{−1}v^{y^{−1}}. Applying π to both sides we conclude w = π(v^{−1})π(v)^{g^{−1}}. Put r = π(v) then w = r^{−1}grg^{−1} and y = wg = r^{−1}gr = g^{r}. Thus (5^{∘}) holds.

We are now in the position to prove the lemma: (a) follows from Lemma 1(b),(e);

(b) from (2^{∘}); (c:a) from (4^{∘}); and (c:b) from (5^{∘}). □

The preceding Lemma allows to control centralizers under the condition y ∉ g^{W}. The next lemma provides us with a tool to achieve this condition: Lemma 3 Let G be a finite group and Π a set of primes. Then there exist a finite abelian ℤG-module W and a monomorphism α : G → W ⋊ G such that

(a) W is a Π-group.

(b) W α(g) = W g for all g ∈ G.

(c) α(g) ∉ g^{W} for all non-trivial Π-elements g in G.

(d) If G is perfect, then W = [W, G] and W ⋊ G is perfect. Proof. Let m be the Π-part of |G|. Put B = (ℤ/mℤ)^{G} and H = ℤ/mℤ ≀ G, where the wreathed product is formed with respect to regular action of G on G. Then B is the base group of H and H = BG. For f ∈ B put ||f|| = ∑_{g∈G} f(g). Put W = [B, G] and note that W = {f ∈ B | ||f|| = 0}. Then W is a Π-group and (a) holds. Also if G = G′, the Three Subgroups Lemma implies [B, G, G] = [B, G] and so W = [W, G] and W G is perfect. Thus (d) holds.

Let b ∈ B be defined by b(1) = 1 and b(g) = 0 for all g ∈ G^{#}. Define α : G → W G, g → g^{b} = [b, g^{−1}]g. Then α is monomorphism and (b) holds. It remains to prove (c). So let g be a non-trivial Π-element in G and suppose that g^{b} = g^{a} for some a ∈ W. Put n = |g| and c = ba^{−1}. Then c ∈ C_{B}(g). Let I be a left transversal to ⟨g⟩. Then each element of G can be uniquely written as ig^{k} for some i ∈ I and some 0 ≤ k < n. Since c^{g} = c, c(i) = c(ig^{k}). Let s = ∑_{i∈I} c(i). We conclude that ||c|| = ns. Thus

1 = ||b|| = ||ac|| = ||a|| + ||c|| = 0 + ns = ns

in ℤ/mℤ, a contradiction, as n divides m. □

We now combine the embeddings from the two preceding lemmas into one: Lemma 4 Let G and F be finite groups and Π a set of primes. Then there exist a finite group G^{∗} with G ≤ G^{∗} and normal subgroup V of G^{∗} such that:

(a) V is an abelian Π-group and G^{∗}/V is simple.

(b) G ∩ V = 1.

(c) Let x be a nontrivial Π-element in G. Then C_{G^{∗}}(x) has a normal solvable Π-subgroup M_{x} with C_{G^{∗}}(x) = M_{x}C_{G}(x).

(d) G^{∗} has a subgroup isomorphic to F.

(e) If G is perfect, G^{∗} is perfect. Proof. Let α and Y = W ⋊ G be as in Lemma 3. Let X be any finite simple group containing Y as a subgroup and such that X has a subgroup isomorphic to F. Let β and V be as in Lemma 2. Put G^{∗} = V ⋊ X. Let x be a Π-element in G and put γ = β ∘ α. Let y ∈ C_{Y} (α(x)). Then W y = W g for some g ∈ G. By Lemma 3(b), W α(x) = W x. From [α(x), y] = 1 we conclude that [x, g] ∈ W ∩ G = 1. Hence g ∈ C_{G}(x) and so C_{Y} (α(x)) ≤ W C_{G}(x) = W α(C_{G}(x)). By Lemma 3(c) α(x) ∉ x^{W} and so by Lemma 2(c:a), C_{G^{∗}}(β(α(x))) ≤ V C_{Y} (α(x)). Thus

C_{G^{∗}}(γ(x)) ≤ V C_{Y} (α(x)) ≤ V W α(C_{G}(x)).

By Lemma 2(b), V β(y) = V y for all y ∈ Y and so V α(C_{G}(x)) = V γ(C_{G}(x)). Thus

C_{G^{∗}}(γ(x)) ≤ V W γ(C_{G}(x)) = V W C_{γ(G)}(γ(x)).

Put M_{x} = C_{V W} (γ(x)). Then C_{G^{∗}}(γ(x)) = M_{x}C_{γ(G)}(γ(x)). Identifying G with its image in G^{∗} under γ we see that all parts of the lemma hold. □

Let G be a locally finite group. Recall that a Kegel-cover for G is a set K of pairs subgroups of G such that

(i) If (H, M) ∈ K, then H is a finite subgroup of G, M ⊴ H and H/M is simple. (ii) For each finite subgroup F of G there exists (H, M) ∈ K with F ≤ H and

F ∩ M. Otto Kegel proved that every locally finite, simple group has a Kegel cover. The

following well-known Lemma is a partial converse: Lemma 5 Let G be a locally finite group with a Kegel cover K. Suppose that for all (H, M) ∈ K, H is perfect and M is solvable. Then G is simple. Proof. Let L be a non-trivial normal subgroup of G. Let g ∈ G. It suffices to show that g ∈ L. For this let 1 ≠ l ∈ L and put F = ⟨l, g⟩. Then F is finite and so there exists (H, M) ∈ K with F ≤ H and F ∩ M = 1. Since l ∈ H \ M we have L ∩ H ≰ M. Since L ∩ H is normal in H and H/M is simple this implies H = (L ∩ H)M. Thus H/L ∩ H ≅ M/M ∩ L and since M is solvable, H/L ∩ H is solvable. As H is perfect we conclude that H = L ∩ H. Thus g ∈ F ≤ H ≤ L. □

Proposition 6 Let G_{0} be a finite, perfect group, Π a non-empty set of primes and for each positive integer n let F_{n} be a finite group. Then there exists a locally, finite simple group G with G_{0} ≤ G and such that:

(a) If x is a nontrivial Π-element of G, then C_{G}(x) has a locally solvable, normal Π-subgroup M_{x} of finite index.

(b) If x is a nontrivial Π-element in G_{0}, then C_{G}(x) = M_{x}C_{G_{0}}(x).

(c) F_{n} is isomorphic to a subgroup of G. Proof. We will produce finite groups G_{n}, n ∈ ℤ^{+}, and normal subgroups M_{n} of G_{n} such that for all n ∈ ℤ^{+} 1^{∘}

(a) G_{n} is perfect.

(b) M_{n} is abelian and G_{n}/M_{n} is simple.

(c) G_{n−1} ≤ G_{n} and G_{n−1} ∩ M_{n} = 1.

(d) If x is a nontrivial Π-element in G_{n−1}, then there exists a solvable normal Π-subgroup M_{nx} of C_{G_{n}}(x) with C_{G_{n}}(x) = M_{nx}C_{G_{n−1}}(x).

(e) G_{n} has a subgroup isomorphic to F_{n}.

Let i ≥ 0 and suppose we already found G_{1}, …, G_{i} such that (a) -(e) hold for 1 ≤ n ≤ i. Then G_{i} is perfect and so we can apply Lemma 4 to G = G_{i} and F = F_{i+1}. Put G_{i+1} = G^{∗}, M_{i} = V and M_{ix} = M_{x}. Then by Lemma 4 we conclude that (a) to (e) hold for n = i + 1.

Put G = ⋃_{i=1}^{n} G_{n}. Then {(G_{n}, M_{n}) | n ≥ 1} is a Kegel cover for G and by Lemma 5, G is simple. Let x ∈ G be a nontrivial Π-element. Then x ∈ G_{n} for some n. Put M_{x}^{n} = 1 and inductively, M_{x}^{m+1} = M_{x}^{m}M_{(m+1)x}. It follows from (1^{∘})(d) and induction that: 2^{∘} Let m ≥ n. Then M_{x}^{m} is a solvable, normal Π-subgroup of C_{G_{m}}(x) and C_{G_{m}}(x) = M_{x}^{m}C_{G_{n}}(x).

Put M_{x} = ⋃_{m=n}^{∞} M_{x}^{m}. Then by (2^{∘}), M_{x} is a locally solvable, normal Π-subgroup of C_{G}(x) and C_{G}(x) = M_{x}C_{G_{n}}(x). Thus the proposition is proved. □

Corollary 7 Let Π be a non-empty set of primes. Then there exists a non-linear, locally finite, simple group G such that

(a) The centralizer of every non-trivial Π-element has a locally solvable Π-subgroup of finite index.

(b) There exists an element whose centralizer is a locally solvable Π-group. Proof. Fix p ∈ Π and put G_{0} = Alt(2p + 1). Let x the product of two disjoint p-cycle in Sym(2p+1). Then x ∈ G_{0}, G_{0} is perfect and C_{G_{0}}(x) ≅ C_{p}×C_{p}. In particular, C_{G_{0}}(x) is solvable Π-group. Apply Lemma 6 to this G_{0} and with F_{n} = Sym(n). The resulting G is not linear and fulfills (a). Moreover, (b) holds for the element x ∈ G_{0} ≤ G. □

Proof of Theorem A:

Apply Corollary 7(a) with Π the set of all primes and Corollary 7(b) with Π = {p}.

References

[1] B. Hartley, Simple locally finite groups in Finite and Locally Finite Groups (B. Hartley et al, Eds.) Kluwer Academic, Dordrecht, 1995, 1-44.

[2] U. Meierfrankenfeld, Lecture Notes on Locally Finite Simple Groups, Michigan State University 1997, http://www.math.msu.edu/∼meier/Classnotes/LFG/LFG abstract.html