CONNECTED COMPONENTS OF THE CATEGORY OF ELEMENTARY ABELIAN p-SUBGROUPS

NADIA MAZZA

Abstract. We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p + 1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1, 2, p + 1, or possibly 3 classes for some 3-groups of maximal nilpotency class.

1. Introduction

The elementary abelian p-subgroups of a finite group are ubiquitous in the study of the p-local structure of the entire group. They provide useful information on the modular representations of the group and its cohomology ring. The main motivation of the author for this research is concerned with the classification of endotrivial modules of a finite group over a field of characteristic p. The analysis in [1, 9, 10] and [11], lead to the definition of the category E_{≥2}(G) of elementary abelian p-subgroups of a finite group G of rank at least 2. It turns out that the torsion-free rank of the group of endotrivial modules for G equals the number n_{G} of connected components of E_{≥2}(G), whence the interest to bound this number. Jon Alperin showed that n_{G} is bounded and depends on p. Note that if n_{G} > 1, then G has a non abelian Sylow p-subgroup with a cyclic center and some maximal elementary abelian p-subgroup of rank 2. Such p-groups have been thoroughly studied by Blackburn ([5]) and in the works involving endo-permutation and endotrivial modules ([1, 6, 7, 9, 10, 11] and [18]).

The objective of this research is to compute an upper bound for n_{G} in the case that G is a finite p-group and that p is odd. Equivalently, this problem can be stated as determining the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2. This will be explained in the next section, which also contains the proof of the main result:

Theorem 2.7. Let p be an odd prime and G a finite p-group. Then n_{G} ≤ p + 1. In particular, if G has rank 2, then there are at most p + 1 conjugacy classes of maximal elementary abelian subgroups of rank 2, and if G has rank at least 3, then there are at most p conjugacy classes of maximal elementary abelian subgroups of rank 2.

In fact, our conclusion is an easy corollary of Héthelyi’s work on soft subgroups ([14, 15]), which builds up essentially from [5]. Let us also mention that Jon Carlson recently handled the case for p = 2, and obtained that n_{G} is at most 5 ([8]). He conjectured that, for p odd, n_{G} ≤ p + 1, which is hence true, as we show in Section 2. We end this note by explictly computing n_{G} and the conjugacy classes of the elementary abelian subgroups of rank 2 in the case that G has rank 2.

2. Generalities: Results old and new

Throughout, p is an odd prime and G denotes a finite p-group. We write H ≤ G (resp. H < G) if H is a subgroup of G (resp. proper subgroup). If g ∈ G, we set ^{g}H = gHg^{−1} and c_{g} for the conjugacy map of G; that is, c_{g}(x) = ^{g}x = gxg^{−1}, for all x ∈ G. For two groups H and K having an isomorphic central subgroup Z, we write H ∗_{Z} K for the central product over Z, or simply H ∗K if there is no confusion. Finally, G′ is the derived subgroup of G, generated by all the commutators [x, y] = x^{−1}y^{−1}xy, for all x, y ∈ G. We refer the reader to [16] for further details.

Definition 2.1. Let G be a finite p-group.

(1) An elementary abelian subgroup of G is an abelian subgroup E of G of exponent p. If E has order p^{a}, then E is a 𝔽_{p}-vector space of dimension a and we call a the rank of E. The rank of G is the maximum of the ranks of all the elementary abelian subgroups of G.

(2) We set E_{≥2}(G) for the category whose objects are the elementary abelian subgroups of G of rank at least 2 and the morphisms are the compositions of conjugations and inclusions.

For short, we write E ∈ E_{≥2}(G) if E is an object of E_{≥2}(G). The definition says that for E, F ∈ E_{≥2}(G), then Hom_{G}(E, F) = {c_{g} | g ∈ G : ^{g}E ≤ F }. In particular, Hom_{G}(E, F) is not empty if and only if E is conjugate to a subgroup of F. We define a connected component of E_{≥2}(G) as follows: E, F ∈ E_{≥2}(G) are connected if there are subgroups E_{0}, …, E_{n} ∈ E_{≥2}(G) with the properties that E_{0} = E, E_{n} = F and, for each 0 ≤ i < n, one of Hom_{G}(E_{i}, E_{i+1}) or Hom_{G}(E_{i+1}, E_{i}) is not empty. We call E ∈ E_{≥2}(G) maximal if the condition Hom_{G}(E, F) ≠ ∅ implies E =_{G} F; that is, E is not properly contained in an elementary abelian subgroup of G. We let n_{G} be the number of connected components of E_{≥2}(G). Recall that we want to find an upper bound for n_{G}. Since n_{G} ≤ 1 whenever G is abelian or has a non cyclic center, let us exclude these cases from now on. Hence, we set Z for the unique central subgroup of G of order p and E_{0} for a normal elementary abelian subgroup of G of rank 2, which exists by [16, Hilfsatz III 7.5]. Define G_{0} = C_{G}(E_{0}). Note that G_{0} is a maximal subgroup of G.

If G has rank 2, then a component of E_{≥2}(G) is a G-conjugacy class of elementary abelian subgroups of rank 2. By [1, § 5], if G has rank 3 or more, then all the objects of E_{≥2}(G) of rank at least 3 are connected and lie in the big component, denoted B. Moreover, E_{0} ∈ B. The remaining components of E_{≥2}(G) are conjugacy classes of maximal elementary abelian subgroups of rank 2, that we call isolated. Hence, E ∈ E_{≥2}(G) is isolated if E is a maximal elementary abelian subgroup of G that does not lie in B. In particular, Z < E and E has rank 2. Note that n_{G} > 1 if and only if G has an isolated subgroup.

Lemma 2.2. Let E ∈ E_{≥2}(G) be isolated and set L = C_{G_{0}}(E).

(1) L is cyclic and E ∩ G_{0} = Z.

(2) N_{G}(E) = L ∗ EE_{0}, with L = Z(N_{G}(E)) and EE_{0} extraspecial of order p^{3} and exponent p.

Proof. Since E is isolated, the intersection L = G_{0} ∩ C_{G}(E) is necessarily cyclic. It follows that C_{G}(E) = L∗E is metacyclic abelian. Moreover, E_{0}◁G implies that Z < E_{0}, whence E∩E_{0} = Z and EE_{0} is non abelian of order p^{3} and exponent p, since E normalises E_{0}. Thus, EE_{0} is extraspecial of order p^{3} and exponent p. In particular, E is normal in L∗EE_{0}, which shows that N_{G}(E) = L∗EE_{0}. Clearly, L = Z(N_{G}(E)). □

Lemma 2.2 has the following immediate consequence.

Corollary 2.3. Let E ∈ E_{≥2}(G) be isolated and assume that E is normal in G. Then G is a central product X ∗_{Z} Z(G), with X extraspecial of order p^{3} and exponent p, and where Z(G) is cyclic. In particular, G has rank 2 and n_{G} = p + 1.

Hence, from now on, we may assume that if E ∈ E_{≥2}(G) is isolated, then N_{G}(E) < G, and thus C_{G}(E) has index at least p^{2} in G. Indeed, by Lemma 2.2, either all isolated subgroups are normal, or none is normal. Then, we appeal to Héthelyi’s work on soft subgroups ([14, 15]), since so are the centralisers of the isolated subgroups.

Definition 2.4. A subgroup A of G is soft if A = C_{G}(A) and |N_{G}(A) : C_{G}(A)| = p.

Let us collect some useful facts on soft subgroups.

Theorem 2.5. [15, § 1, 2 and 3] Let G be a finite p-group, and let A be a soft subgroup of G.

(1) The subgroups containing A form a chain A = N_{0}, N_{1}, …, N_{k} = M, where N_{i} = N_{G}(N_{i−1}), for 1 ≤ i ≤ r, and |G : M| = p.

(2) M is the unique maximal subgroup of G containing A, and M has nilpotency class k + 1.

(3) Any two soft subgroups of G contained in M are G-conjugate.

(4) M has exactly p maximal subgroups containing a soft subgroup of G.

(5) If |G : A| ≥ p^{2}, then H = G′Z(N_{G}(A)) is independent of A and the factor group G/H is elementary abelian of order p^{2}. Moreover, for x ∈ H, then the subgroup C_{G}(x) of G is not soft, or |G : C_{G}(x)| < p^{2}. The conjugates of A are the subgroups C_{G}(s), for s ∈ M − H.

As an easy corollary of Theorem 2.5, we get the following.

Proposition 2.6. Assume that G has some isolated subgroup which is not normal in G. Then, 2 ≤ n_{G} ≤ p + 1 and G has at most p maximal elementary abelian subgroups of rank 2.

Proof. Clearly, n_{G} > 1. Let E ∈ E_{≥2}(G) be isolated. Set A = C_{G}(E) and H = G′Z(N_{G}(A)). Then, A satisfies the hypothesis of Theorem 2.5. Thus, H is a normal subgroup of G, independent of the choice of E, and G/H is elementary abelian of rank 2. In particular, G has p + 1 maximal subgroups that contain H and hence might contain some isolated subgroup of G. Now, H < G_{0}, since Z(N_{G}(A)) centralizes E_{0} and since G′ is contained in any maximal subgroup of G. Since G_{0} does not contain any isolated subgroup of G, there are at most p maximal subgroups containing an isolated subgroup, whence the claim by Theorem 2.5 (3). □

Observe that Theorem 2.5 provides more information on the structure of G than the fact deducted in Proposition 2.6. In particular, under the same assumptions, the nilpotency class of G is n − r and that of H is n − r − 1, where r is defined by |Z(N_{G}(E))| = p^{r}. However, for the purpose of this research, let us focus on the consequences of Corollary 2.2 and Proposition 2.6.

Theorem 2.7. Let p be an odd prime and G a finite p-group. Then n_{G} ≤ p + 1. In particular, if G has rank 2, then there are at most p + 1 conjugacy classes of maximal elementary abelian subgroups of rank 2, and if G has rank at least 3, then there are at most p conjugacy classes of maximal elementary abelian subgroups of rank 2.

Corollary 2.8. The torsion-free rank of the group of endotrivial modules of a finite p-group, for an odd prime p, is at most p + 1.

We refer the reader to [8, 9] for further details on Corollary 2.8. We end this section with a criteria stating when n_{G} = 1.

Proposition 2.9. Let G be a finite p-group, p odd, and let V be a normal elementary abelian subgroup of rank r, with r > p. Then n_{G} = 1.

Proof. We need to show that any E ∈ E_{≥2}(G) is contained in an elementary abelian subgroup F of rank at least 3. Let E ∈ E_{≥2}(G) have rank 2. If E does not contain a central subgroup Z of order p, then we set F = EZ. Otherwise, consider V as 𝔽_{p}-vector space. The conjugation action of G induces a map E → Aut V. Via the choice of a basis for V, we have that Aut V ≅ SL_{r}(p) and any x ∈ E identifies with a matrix m_{x} in SL_{r}(p) with order 1 or p. If m_{x} = 1, for all x ∈ E, then E ≤ C_{G}(V), whence E < V E = F, which is elementary abelian of rank at least 3. Otherwise, there is x ∈ E with x ∉ V, and E = Z × ⟨x⟩, with Z = V ∩ E. Then, m_{x} has two non collinear eigenvectors v, w ∈ V, since m_{x} is an element of SL_{r}(p) with order p. That is, x ∈ C_{G}(⟨v, w⟩), and hence F = E⟨v, w⟩ ∈ E_{≥2}(G) has rank three (note that Z < ⟨v, w⟩). □

Let us point out that Proposition 2.9 is in fact a trivial consequence of a more general result of Y. Berkovich ([3, Proposition 20]). This discussion naively leads us to the question: “Let G be a p-group of rank at least p + 1, for an odd prime p. Is n_{G} = 1?”

The answer is not yet known to the author, except in case p = 3.

Proposition 2.10. If p = 3 and G is a 3-group of rank at least 4, then n_{G} = 1.

Proof. By [17, Theorem], if G contains an elementary abelian subgroup of rank 4, then G contains a normal elementary abelian subgroup of the same rank. The result follows by Proposition 2.9. □

If p ≥ 5, then a p-group of rank r > p does not contain any normal elementary abelian subgroup of rank r in general, as shown by Professor Glauberman ([13]). Note that the same conclusion is suggested by [2, Theorem A]. However, in the examples considered, it is always true that if G has rank at least p + 1, then n_{G} = 1.

3. p-groups of rank 2

Throughout this section, we let G be a finite p-group of rank 2, for an odd prime p. We compute n_{G} and describe the conjugacy classes of elementary abelian subgroups of G of rank 2, that is, the connected components of E_{≥2}(G). Let us recall the classification of the p-groups established by Blackburn ([4]). For convenience, we use the notation of [12].

There are four types of p-groups of rank 2 ([12, Theorem A.1]):

• The non-cyclic metacyclic groups M(p, r), of order p^{r}, r ≥ 2.

• The groups C(p, r), of order p^{r}, r ≥ 3, defined by

C(p, r) = ⟨a, b, c | a^{p} = b^{p} = c^{p^{r−2}} = [a, c] = [b, c] = 1, [a, b] = c^{p^{r−3}} ⟩ .

• The groups G(p, r, ε), with ε = 1 or ε is not congruent to a square modulo p, of order p^{r}, r ≥ 4, defined by

G(p, r, ε) = ⟨a, b, c | a^{p} = b^{p} = c^{p^{r−2}} = [b, c] = 1, [a, b^{−1}] = c^{εp^{r−3}} , [a, c] = b ⟩ .

• If p = 3, the following 3-groups of maximal nilpotency class of the form B(3, r; β, γ, δ), of order 3^{r}, r ≥ 4. Fix a set {s, s_{1}, …, s_{r−1}} of generators of G, subject to the relations:

(1) s_{i} = [s_{i−1}, s], 2 ≤ i ≤ r − 1 ;

(2) [s_{1}, s_{2}] = s_{r−1}^{β} ;

(3) [s_{1}, s_{i}] = 1, 3 ≤ i ≤ r − 1 ; (4) s^{3} = s_{r−1}^{δ} ;

(5) s_{1}^{3}s_{2}^{3}s_{3} = s_{r−1}^{γ}

(6) s_{i}^{3}s_{i+1}^{3}s_{i+2} = 1, 2 ≤ i ≤ r − 1 and with s_{r} = s_{r+1} = 1 .

Hence, the groups of maximal nilpotency class are given by the sets of parameters (β, γ, δ) taking the values:

– For r ≥ 5, (β, γ, δ) = (1, 0, δ) and δ ∈ {0, 1, 2} ;

– For even r ≥ 4, (β, γ, δ) ∈ {(0, 1, 0), (0, 2, 0), (0, 0, 0), (0, 0, 1)} , except B(3, 4; 0, 1, 0);

– For odd r ≥ 5, (β, γ, δ) ∈ {(0, 1, 0), (0, 0, 0), (0, 0, 1)} . In particular, we have isomorphisms G(3, 4, 1) ≅ B(3, 4; 0, 0, 0) and G(3, 4, −1) ≅ B(3, 4; 0, 2, 0).

The group B(3, 4; 0, 1, 0) is the wreath product C_{3} ≀ C_{3}, which has rank 3.

3.1. The non-cyclic metacyclic groups M(p, r), r ≥ 3.

If G = M(p, r), then G has a unique elementary abelian subgroup of rank 2, by [19, Lemma 2.1].

Proposition 3.1. If G = C(p, r), then n_{G} = 1 and E_{≥2}(G) = B = {E_{0}}.

3.2. The groups C(p, r), r ≥ 3.

By [12, Lemma A.5], C(p, r) is a non-split central extension

\xymatrix{1 \ar[r] & Z(C(p, r)) \ar[r] & C(p, r) \ar[r] & C(p, 3) \ar[r] & 1 ,}

where Z(C(p, r)) = ⟨c⟩ is cyclic of order p^{r−2}. In addition, C(p, 3) is an extraspecial group of order p^{3} and exponent p and contains all the elements of order p of C(p, r). Thus, we fall into the situation of Corollary 2.3, and so any E ∈ E_{≥2}(C(p, r)) is normal in C(p, r).

Proposition 3.2. If G = C(p, r), then n_{G} = p + 1, and each component of E_{≥2}(G) is a single subgroup.

3.3. The groups G(p, r, ε), r ≥ 4 and ε = 1 or ε is not congruent to a square modulo p.

Write G = G(p, r, ε) = ⟨a, b, c | a^{p} = b^{p} = c^{p^{r−2}} = [b, c] = 1, [a, b^{−1}] = c^{εp^{r−3}} , [a, c] = b ⟩ .

We have

\xymatrix{1 \ar[r] & ⟨b, c⟩ \ar[r] & G \ar[r] & ⟨a⟩ \ar[r] & 1 ,}

and z = c^{p^{r−3}} generates the unique central subgroup of order p. Hence, we can choose E_{0} = ⟨b, z⟩, and the other objects of E_{≥2}(G) are E_{i} = ⟨ab^{i}, z⟩ , 1 ≤ i ≤ p. Indeed, it follows from [12, Lemma A.6], or from a routine computation, that the elements of order p of G − E_{0} are of the form a^{i}b^{j}z^{k}, with 0 ≤ i, j, k ≤ p − 1. Moreover, ^{c}E_{i} = E_{i−1}, where the index is taken modulo p, in the range 1, …, p. Thus {E_{i} | 1 ≤ i ≤ p} is a single conjugacy class, i.e. a component of E_{≥2}(G).

Proposition 3.3. If G = G(p, r, ε), then n_{G} = 2, with components B = {E_{0}} and {E_{i} | 1 ≤ i ≤ p}.

3.4. The 3-groups of maximal nilpotency class B(3, r; β, γ, δ), r ≥ 4.

In addition to the above notation, let us set G_{i} = ⟨s_{i}, …, s_{r−1}⟩, for all 1 ≤ i ≤ r − 1. Then, G ▷ G_{2} ▷ G_{3} ▷ ⋯ ▷ G_{r−1} is the lower central series of G and G_{r−1} = Z(G) has order p. Moreover, G_{1} = ⟨s_{1} , s_{2}⟩ is metacyclic non cyclic, of index p in G, and G_{1} is abelian if and only if β = 0. In this case, if r = 2k, then G_{1} = ⟨s_{1}⟩ × ⟨s_{2}⟩ ≅ C_{k} × C_{k−1} and if r = 2k + 1, then G_{1} = ⟨s_{1}⟩ × ⟨s_{2}⟩ ≅ C_{k} × C_{k}. Also, if ⟨s̄⟩ = G/G_{1} ≅ C_{3}, then the extension

\xymatrix{1 \ar[r] & G_{1} \ar[r] & G \ar[r] & ⟨s̄⟩ \ar[r] & 1}

splits if and only if δ = 0.

In any case, G_{1} has a unique elementary abelian subgroup of rank 2, which is hence characteristic in G. Clearly, this subgroup is G_{r−2} = Ω_{1}(G_{1}), and we have C_{G}(G_{r−2}) = G_{1}.

Now, we want to determine the conjugacy classes of the elements E ∈ E_{≥2}(G), with E ≠ G_{r−2}. Therefore, we first find the elements of order 3. Observe that any element x ∈ G can be uniquely written as x = s^{ζ}s_{1}^{ζ_{1}} ⋯ s_{r−1}^{ζ_{r−1}}, for integers ζ, ζ_{i} ∈ {−1, 0, 1}, for all 1 ≤ i ≤ r − 1. From [12, Proposition A.9], we obtain that the elements of order 3 in G are either in G_{r−2} or belong to a coset of G/G_{2} satisfying ζ(δ + βζ_{1}^{2}) + γζ_{1} ≡ 0 mod 3 , with ζ = ±1 . The solutions for ζ_{1} are:

r  (β, γ, δ)  ζ_{1}
r ≥ 4 (0, 1, 0)  0
r ≥ 4 (0, 0, 0)  −1, 0, 1
r ≥ 4 (0, 0, 1) no solution
r even (0, 2, 0)  0
r ≥ 5 (1, 0, 0)  0
r ≥ 5 (1, 0, 1) no solution
r ≥ 5 (1, 0, 2)  −1, 1

From this table, we immediately deduce the following.

Proposition 3.4. If δ = 1, then n_{G} = 1 and E_{≥2}(G) = B = {G_{r−2}} .

Assume now that δ = 0, and that G ≅ B̸(3, 4; 0, 1, 0). Then, the extension

\xymatrix{1 \ar[r] & G_{1} \ar[r] & G \ar[r] & ⟨s̄⟩ \ar[r] & 1}

splits, and we have that Ω_{1}(G) = G_{r−2} ⋊ ⟨s⟩ is extraspecial of order 27 and exponent 3. Thus, except G_{r−2}, the group G has 3 elementary abelian subgroups E_{i} of rank 2, generated by a central element z of order 3 and x_{i} = ss_{r−2}^{i}, for i ∈ {1, 2, 3}. Moreover, E_{1}, E_{2} and E_{3} are conjugate.

Proposition 3.5. If δ = 0 and G ≅ B̸(3, 4; 0, 1, 0), then n_{G} = 2, with components B = {G_{r−2}} and {E_{1}, E_{2}, E_{3}}.

Proposition 3.3 handles the cases B(3, 4; 0, 0, 0) ≅ G(3, 4, 1) and B(3, 4; 0, 2, 0) ≅ G(3, 4, −1), whereas B(3, 4; 0, 1, 0) has rank 3. Hence, we are left with the case that r ≥ 5 and (β, γ, δ) = (1, 0, 2). By the above discussion, the elements of order 3 that do not belong to G_{1} are all the elements of the cosets of G/G_{2} of the form s^{±1}s_{1}^{±1}G_{2}. Set x = ss_{1}, z = s_{r−1} and let E = ⟨x, z⟩ =

{1, x = ss_{1} , x^{2} = s^{2}s_{1}^{2}s_{2}z^{−1} , z , xz = ss_{1}z , x^{2}z = s^{2}s_{1}^{2}s_{2} , z^{2} , xz^{2} = ss_{1}z^{2} , x^{2}z^{2} = s^{2}s_{1}^{2}s_{2}z} .

Since x ∉ G_{1}, [16, Hilfsatz III.14.3] shows that E = C_{G}(E), and so, the conjugacy class of x is the coset xG_{2} ⊂ G/G_{2}. Notice also that N_{G}(E) = EG_{r−2} is extraspecial of order 27 and exponent 3. It follows that there are exactly two conjugacy classes of elementary abelian subgroups that are not normal in G.

Proposition 3.6. If δ = 2, then n_{G} = 3, with components B = {G_{r−2}} and the conjugacy classes of ⟨ss_{1}, s_{r−1}⟩ and of ⟨ss_{1}^{−1}, s_{r−1}⟩.

ACKNOWLEDGMENTS

The author is sincerely grateful to Dave Benson, Jon Carlson, Antonio Diaz, George Glauberman, Adam Glesser, Geoff Robinson and Jacques Thévenaz for their helpful advices and comments. The author also wish to thanks the Swiss National Foundation for its financial support.

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