MAXIMAL ELEMENTARY ABELIAN SUBGROUPS OF RANK 2

JON F. CARLSON^{∗}

1. Introduction

The poset A of all elementary abelian p-subgroups of a finite group or of all psubgroups of a finite group plays a significant role in the modular representation theory and cohomology of the group. The posets and associated complexes have been used for the computation of cohomology as well as a setting for major problems such as Alperin’s conjecture. See one of the texts [3] or [8] for further discussion. In this paper we consider a variation on this theme. We suppose that G is a finite group and let P = P(G) denote the partially ordered set of all elementary abelian 2-subgroups of G having rank at least 2. The ordering on P is ordinary inclusion of subgroups. Note that P has an action of G by conjugation. So it makes sense to consider the orbit space P/G. Our main theorem is the following.

Theorem 1.1. The orbit space P/G has at most 5 connected components.

The poset P is of interest beyond its obvious connection to A. For one thing, the number of components is the dimension of H_{0}(P/G) which is a part of the obstruction to solving the gluing problem in block theory for certain groups [5]. But also the number of components for P/G is the rank of the group of endotrivial kG-modules in characteristic p. This fact was proved by Alperin [1] in the case that G is a p-group and is proved in [6, 7] for the general case. Specifically, we have the following.

Corollary 1.2. Suppose that k is a field of characteristic 2 and that G is a finite group. Then the torsion free rank of T (G), the group of endotrivial kG-modules, is at most 5.

The proof of the theorem is almost entirely group theory. The point is that there is a theorem of Alperin [1] which says that all elementary abelian subgroups of rank 3 are in the same connected component of P/G. Hence, counting the number of components is primarily an exercise in counting the number of conjugacy classes of maximal elementary abelian subgroup of rank 2 in G. That is, the class of an elementary abelian subgroup of rank 2 is an isolated point in P/G and there is at most one component other than these isolated points. Thus, the proof reduces easily to the case that G is a 2-group, and the theorem that we prove is the following.

Theorem 1.3. Suppose that G is a 2-group. If the 2-rank of G is 3 or more, then there are at most 4 conjugacy classes of maximal elementary abelian subgroups of order 4. If the 2-rank of G is 2 then there are at most 5 conjugacy classes of maximal elementary abelian subgroups of order 4.

Date: January 18, 2006.

∗ This research was supported in part by a grant from NSF.

For the proof we use a recent result of Janko [9] on 2-groups which have no normal subgroup of order 8. In the next section we prove some basic results on 2groups generated by involutions. These are fit into the scheme of Janko’s Theorem in Section 3 and yield a proof of the Theorem 1.1. Some remarks on the odd characteristic case and other related results are presented in the concluding section.

The author wants to thank the Centre Bernoulli and the École Polytechnique Fédérale de Lausanne for their hospitality and support during the time when these theorems were discovered. Special thanks are due to Jon Alperin for directing us to Janko’s paper and to Jacques Thévenaz for many helpful conversations. We were led to conjecture the main theorems by experimental evidence obtained using the computer algebra system MAGMA [4].

2. Groups generated by involutions

In this section we investigate some problems involving 2-groups that have maximal elementary abelian subgroups of rank 2 and are generated by involutions. Our main result is the following. One thing to remember is that if x and y are involutions in a group G, then ⟨x, y⟩ is a dihedral group. For notation, we let o(x) denote the order of the element x. In particular, if o(x) is a power of two that is at least 4, then the element x^{o(x)/4} has order exactly 4.

Proposition 2.1. Suppose that G is a finite 2-group and that x_{1}, x_{2} and x_{3} are distinct involutions with the following properties:

(1) G = ⟨x_{1}.x_{2}, x_{3}⟩;

(2) for i = 1, 2, 3, C_{G}(x_{i}) has rank 2;

(3) if both x_{2} and x_{3} do not commute with x_{1}, then

⟨(x_{1}x_{2})^{o(x_{1}x_{2})/4}⟩ = ⟨(x_{1}x_{3})^{o(x_{1}x_{3})/4}⟩.

Then G is a dihedral group.

Proof. First note that if G is abelian then by (2), G is elementary abelian of order 4. Thus the conclusion is satisfied assuming the usual convention that an elementary abelian group of order 4 is dihedral.

So suppose that both x_{2} and x_{3} commute with x_{1}. Then we can assume that x_{2} and x_{3} do not commute and that x_{1} is not an element of ⟨x_{2}, x_{3}⟩. Also we have that G ≅ C_{G}(x_{1}). Let z be an element of order two in the center of ⟨x_{2}, x_{3}⟩. Then ⟨z, x_{1}, x_{2}⟩ is an elementary abelian 2-group in C_{G}(x_{2}) of rank 3, contradicting hypothesis (2).

Now suppose that x_{1} commutes with x_{3} but not with x_{2}. Then H = ⟨x_{1}, x_{2}⟩ is a dihedral group of order at least 8. By condition (2), C_{G}(x_{1}) has a unique elementary abelian subgroup which is generated by x_{1} and the unique involution in the center of G. But this subgroup must also contain x_{3} and it is contained in H. So G = H is dihedral as asserted.

We assume from here on that x_{1} does not commute with either x_{2} or x_{3}. We can also suppose that x_{2}x_{3} ≠ x_{3}x_{2}, as otherwise, we can argue as in the previous paragraph with the roles of x_{1} and x_{2} interchanged.

We proceed by induction on |G|. Note that the smallest groups satisfying the conditions of the theorem are the elementary abelian group of order 4 and the dihedral group of order 8. Both of these groups satisfy the conclusion of the theorem. We further assume that G is not a dihedral group. From this it follows that G/ϕ(G) ≅ (ℤ/2ℤ)^{3} where ϕ(G) is the Frattini subgroup of G. The reason is that if |G/ϕ(G)| were 4, then G would be generated by two of the elements x_{1}, x_{2}, x_{3} and would be a dihedral group. Let z denote the unique involution in the center of G. Then z must be an element of the center of ⟨x_{1}, x_{2}⟩ and z must be in the Frattini subgroup.

Now let u = (x_{1}x_{2})^{o(x_{1}x_{2})/4} and v = (x_{2}x_{3})^{o(x_{2}x_{3})/4}. Because u^{2} and v^{2} commute with x_{2} it must be that u^{2} = v^{2} is the involution z in the center of G. That is, by condition (2), the subgroup W = C_{G}(x_{2}) has a unique elementary abelian subgroup of order 4. It is the subgroup ⟨x_{2}, z⟩, and it is central in C_{G}(x_{2}). But u^{2} and v^{2} are in W ∩ ϕ(G) = ⟨z⟩.

Next we claim that u and v commute. This is because conjugation by both x_{2} and x_{3} invert the element u by condition (3). Thus x_{2}x_{3} centralizes u, and hence v also centralizes u.

So (uv)^{2} = 1 and uv commutes with x_{2}. Consequently, uv is in the subgroup W = ⟨z, x_{2}⟩. We claim that uv cannot be equal to x_{2} or to x_{2}z. The reason is that the latter two elements are both congruent to x_{2} modulo the Frattini subgroup and there is no way that uv = (x_{1}x_{2})^{o(x_{1}x_{2})/4}(x_{2}x_{3})^{o(x_{2}x_{3})/4} can be congruent to x_{2} modulo ϕ(G). Therefore we have either that uv = z or that uv = 1. In either case, ⟨u⟩ = ⟨v⟩.

We have proved at this point that condition (3) of the theorem holds for any permutation of the indices 1, 2 and 3. Hence the following further assumption is justified. Assume that x_{1} and x_{2} are chosen so that o(x_{1}x_{2}) is maximal in the set {o(x_{1}x_{2}), o(x_{1}x_{3}), o(x_{2}x_{3})}

Let K_{1} = ⟨x_{1}, x_{3}x_{1}x_{3}, x_{2}⟩, and notice that K_{1} satisfies all four conditions in the hypothesis of the theorem. So, by induction, K_{1} is a dihedral group. Because of the maximality of o(x_{1}x_{2}), we have that K_{1} = ⟨x_{1}, x_{2}⟩. For the same reason, K_{2} = ⟨x_{1}, x_{2}, x_{3}x_{2}x_{3}⟩ = ⟨x_{1}, x_{2}⟩. As a consequence the subgroup H = ⟨x_{1}, x_{2}⟩ contains both x_{3}x_{1}x_{3} and x_{3}x_{2}x_{3} and is normalized by x_{3}. Thus H is a maximal subgroup in G and we have an exact sequence

\xymatrix{1 \ar[r] & H \ar[r] & G \ar[r] & ⟨x_{3}⟩ \ar[r] & 1}

which is split. It follows that G is determined entirely by the action of x_{3} on H.

For convenience of notation, let x = x_{1}, y = x_{1}x_{2} and w = x_{3}. Then H = ⟨x, y⟩, and the subgroup ⟨y⟩ is characteristic in H. For some integers s and t we have that

wxw = xy^{s}
wyw = y^{t}

Clearly, t is an odd number. Let o(y) = 2^{n} and m = 2^{n−1}. So |H| = 4m. Because w^{2} = 1, we have that t is one of 1, −1, m + 1 or m − 1. Also we know that wy^{m/2}w = y^{−m/2}. That is, x_{3} inverts (x_{1}x_{2})^{o(x_{1}x_{2})/4} by Condition (3). So, in fact, either t = −1 or t = m − 1

We claim that s is even. For if s is not divisible by 2, then

(wx)^{2} = wxwx = xy^{s}x = y^{−s}

is a generator for the subgroup ⟨y⟩. In this case G = ⟨x, wx⟩ is a dihedral group. Now for any integer r consider the product

(wxy^{r})^{2} =(wxw)(wy^{r}w)xy^{r} = xy^{s}y^{tr}xy^{r}
=y^{−s−tr}y^{r} = y^{−s−r(t−1)}
= \begin{cases}y^{2r−s} & if t = −1 \\ y^{r(m−2)−s} & if t = m − 1 \end{cases}

Notice that in the second case m − 2 = 2^{n−1} − 2 = 2(2^{n−2} − 1) ≡ 2 mod 4. Because s is even, it is possible in either case to choose r so that (wxy^{r})^{2} = 1.

The result of this is that

((wy^{1−r})(y^{−1}x))^{2} = 1,

since xy^{r} = y^{−r}x. If either t = −1 or r − 1 is an even integer, then wy^{r−1} is an involution. That is, if t = −1 then w inverts y while if t = m − 1 then w inverts y^{2}. So, in either of these cases wy^{r−1} commutes with the involution y^{−1}x = xy, and the subgroup ⟨wy^{r−1}, xy, y^{m}⟩ is elementary abelian of order 8. This is a contradiction to the hypothesis that the centralizer of xy ( = x_{2} in the original notation) has rank 2.

So we are left with the case that t = m − 1 and that r − 1 is odd. Hence, r is even and by the same formula

((wy^{r})(x))^{2} = 1.

In this case, wy^{r} is an involution, and ⟨wy^{r}, x, y^{m}⟩ is an elementary abelian subgroup of order 8 in the centralizer of x = x_{1}. Again we have a contradiction. Hence the theorem is proved. □

For the proof of our main theorem in the next section, we need the following consequence. Recall that for G a p-group, Ω_{n}(G) denotes the subgroup generated by all elements of order p^{n}.

Proposition 2.2. Suppose that G is a 2-group with a maximal subgroup H such that Ω_{2}(H) is abelian of type (4, 2) or of type (4, 4). Let u ∈ G be an element that is not in H. There are at most two conjugacy classes of involutions x in uH such that the rank of C_{G}(x) equal to 2.

Proof. Suppose that x_{1}, x_{2} and x_{3} are distinct involutions in uH with the property that C_{G}(x_{i}) has rank 2 for i = 1, 2, 3. Assume further that no two of x_{1}, x_{2} or x_{3} are conjugate in G. Notice that in this case, the center of G is cyclic, since otherwise, there can not exist three distinct involutions not in H whose centralizers all have rank 2. Let W = Ω_{2}(H) and let W_{0} = Ω_{1}(H) ⊆ W be the unique elementary abelian subgroup of order 4 in H. Then W_{0} = ⟨z, y⟩ where z is the unique involution in the center of G.

Note that x_{1}, x_{2} and x_{3} can not all commute since otherwise, x_{1}, x_{2}, x_{3} and z would be four distinct involutions in the centralizer of x_{1}. Now suppose that x_{1} commutes with x_{2} but not with x_{3}. Then x_{2} ∈ Ω_{1}(C_{G}(x_{1})) = ⟨x_{1}, z⟩. On the other hand, Ω_{1}(C_{G}(x_{1})) is a subgroup of ⟨x_{1}, x_{3}⟩, which is a dihedral group. So x_{2} must be conjugate to either x_{1} or x_{3}. Hence, it must be that no two of x_{1}, x_{2} and x_{3} commute.

For each i, x_{i} does not centralize W_{0}. Hence it must be that x_{i}yx_{i} = zy. For i ≠ j, x_{i} and x_{j} do not commute, and hence ⟨x_{i}, x_{j}⟩ is a dihedral group with x_{i}x_{j} ∈ H. Let u_{i} = (x_{1}x_{i})^{o(x_{1}x_{i})/4} for i = 2, 3. Note that x_{1} inverts both u_{2} and u_{3}.

Now consider the conjugation action of x_{1} on W. Suppose first that W is of type (4, 4). Then W = ⟨z_{1}, y_{1}⟩ where z_{1}^{2} = z and y_{1}^{2} = y. Because the squaring operation induces a G-invariant homomorphism \xymatrix{W/W_{0} \ar[r] & W_{0}} it must be that x_{1}y_{1}x_{1} ≡ y_{1}z_{1} modulo W_{0}. Hence, the only cyclic subgroup of order 4 that is stabilized by conjugation by x_{1} is generated by an element of the form z_{1}a for a some element of W_{0}. Thus ⟨z_{1}⟩ and ⟨z_{1}y⟩ are the only two cyclic subgroup of order 4 that can be stabilized by x_{1}. If conjugation by x_{1} inverts z_{1}, then x_{1}z_{1}x_{1} = z_{1}z and x_{1}z_{1}yx_{1} = z_{1}y. That is, if conjugation by x_{1} inverts z_{1}, then it does not invert z_{1}y. Consequently, there is a unique cyclic subgroup of order 4 in H with the property that x_{1} inverts the generator of that group. This subgroup is ⟨u_{2}⟩ = ⟨u_{3}⟩.

The same situation is encountered in the case that W is of type (4, 2). In this case, W = ⟨z_{1}, y⟩, where z_{1}^{2} = z, since ⟨z⟩ is the unique G-invariant subgroup of order 2 of W. Again, there are two cyclic subgroups of order 4: ⟨z_{1}⟩ and ⟨z_{1}y⟩. Again, x_{1}yx_{1} = zy, and if conjugation by x_{1} inverts z_{1} then it fixes z_{1}y. So there is a unique subgroup of H which is cyclic of order 4 and whose generator is inverted by x_{1}. Again it is ⟨u_{2}⟩ = ⟨u_{3}⟩.

To this point we have proved that the group T = ⟨x_{1}, x_{2}, x_{3}⟩ satisfies all of the hypotheses of Theorem 2.1. Therefore, T is a dihedral group and T has exactly two conjugacy classes of noncentral involutions. It follows that two of the elements x_{1}, x_{2} and x_{3} are conjugate. This proves the theorem. □

3. Proof of the main theorems

In order to prove Theorems 1.1 and 1.3 we need to draw on some known facts about 2-groups. The primary result that we need is the theorem of Janko quoted with only minor paraphrase below.

Theorem 3.1. [9] Let G be a finite 2-group which does not have a normal elementary abelian subgroup of order 8. Suppose that G is neither abelian nor of maximal class. Then G has a normal metacyclic subgroup N such that C_{G}(Ω_{2}(N)) ≤ N and one of the following holds:

(1) |G/N| ≤ 4 and Ω_{2}(N) is abelian of type (4, 4),

(2) |G/N| ≤ 4 and N is abelian of type (2^{ℓ}, 2) for ℓ ≥ 2, (3) G/N ≅ D_{8} and N is abelian either of type (2^{ℓ}, 2^{ℓ+1}) for ℓ ≥ 2, or of type

(2^{ℓ}, 2^{ℓ}) with ℓ ≥ 2,

(4) G/N ≅ D_{8} and N is minimal nonabelian, Ω_{2}(N) is abelian of type (4, 4), and more precisely, N = ⟨a, b|a^{2^{m}} = b^{2^{n}} = 1, a^{b} = a^{1+2^{m−1}}⟩, where m = n with n ≥ 3 or m = n + 1 with n ≥ 2.

Now we begin the proof of Theorem 1.3. We assume that G is a 2-group with a maximal elementary abelian subgroup of order 4 and with at least two conjugacy classes of maximal elementary abelian subgroups. Assume also that G is not a dihedral group. Let z be the unique involution in the center of G. The first thing we need is the following.

Lemma 3.2. The group G has no normal elementary abelian subgroup of order 8.

Proof. Suppose that E is a normal elementary abelian subgroup of order 8 in G and that F is a maximal elementary abelian subgroup of order 4. Let y be a noncentral involution in F. Then F = ⟨y, z⟩ is the unique maximal elementary abelian subgroup of C_{G}(y). Now y acts on E by conjugation. and because y has order 2, the fixed subgroup of the action of y on E must be of order 4 or larger. Hence E ∩ C_{G}(y) has order at least 4, which is a contradiction. □

For notation, let N be the metacyclic subgroup of G that is guaranteed by Theorem 3.1. Let W = Ω_{2}(N) and let W_{0} = Ω_{1}(N) be the unique maximal elementary abelian subgroup of N. In particular, N contains at most one conjugacy class of maximal elementary abelian subgroups of order 4. Let W_{0} = ⟨y, z⟩.

Lemma 3.3. Suppose that E is a maximal elementary abelian subgroup of order 4 which is not contained in N. Then E = ⟨u, z⟩ for some involution u ∉ N. Moreover, there are at most two conjugacy classes of maximal elementary abelian subgroups of order 4 in H = ⟨u, N⟩ that are not in N.

Proof. We need only notice that N is a subgroup of index 2 in H. The rest is a direct consequence of Theorem 2.2. □

We can now settle most of the proof of Theorem 1.3.

Lemma 3.4. If, for the group G, conditions (b), (c) or (d) of Theorem 3.1 hold, then the conclusion of Theorem 1.3 is valid.

Proof. The point of Lemma 3.3 is that we need to count the number of cosets uN where uN is an involution in G/N and uN has the possibility of containing an involution w such that ⟨w, z⟩ is a maximal elementary abelian subgroup. Now notice that if uN and vN are two such cosets which are conjugate in G/N then uN ∪ vN has at most two classes of involutions x such that the rank of C_{G}(x) is two (see 2.2). Therefore, we need only count one of the cosets uN or vN.

In addition, in every one of the cases (b), (c) and (d) of Theorem 3.1, the subgroup W_{0} is central in N and normal in G. If u ∉ N is an involution such that ⟨u, z⟩ is a maximal elementary abelian subgroup, then u can not centralize W_{0}. Specifically, it must be that uyu = yz. For this reason, uN can not be in the Frattini subgroup of G/N and if uN and vN are two cosets which contain involutions whose centralizers have rank 2, then uvN can not contain any such involution. The result of all of this is that in every one of the cases (b), (c) and (d) of Theorem 3.1, there are at most two cosets that can contribute conjugacy classes of involutions whose centralizers have rank 2. By Theorem 1.3 there are at most four conjugacy classes of maximal elementary abelian subgroups of order 4 that are not contained in N. □

Proof of Theorem 1.3. To finish the proof of Theorem 1.3 we need only deal with case (a) in Theorem 3.1. Unfortunately, the statement of the theorem is not explicit enough for our purposes, and we must actually delve into the details of Janko’s proof. Specifically, we must take a detailed look at some of the cases that are considered in the proof which runs from page 956 to 961 of [9]. Most of this is handled very easily, as several of the cases result in group structure as expressed in parts (b), (c) or (d) of Theorem 3.1. For example, we point out that the subgroups F, H, K and L (pages 958-959) all centralize W_{0} and hence our proof follows along the lines of Lemma 3.4.

The only difficulties come with the analysis on page 959 of [9]. The case (A) on that page is not a problem, because the automorphism group denoted A_{0} fixes W_{0} point wise. Consequently, it is impossible to have a involution not in C = C_{G}(W) whose centralizer has rank 2. The case (C) on page 959 is shown not to exist.

So there remains only case (B) on page 959. In this case, C = C_{G}(W), W = Ω_{2}(C) and G/C has order 16. Moreover, G/C acts on W by the group of automorphisms (in the notation of [9])

⟨ρ, σ, η⟩ = ⟨ρ, σ⟩ × ⟨η⟩ ≅ D_{8} × C_{2}

where

ρ(y_{1}) = y_{1}^{−1}, ρ(z_{1}) = z_{1}, σ(y_{1}) = y_{1}z_{1}, σ(z_{1}) = z_{1}, η(y_{1}) = y_{1}^{−1}, η(z_{1}) = z_{1}^{−1}

for W = ⟨y_{1}, z_{1}⟩, y_{1}^{2} = y and z_{1}^{2} = z. Again we are looking for involutions in G/C which act nontrivially on W_{0}. Note that η and ρ both centralize W_{0}. So the only involutions which act nontrivially on W_{0} are σρ, σ^{−1}ρ, σρη, and σ^{−1}ρη. Because σρ is conjugate to σ^{−1}ρ and σρη is conjugate to σ^{−1}ρη, we have that there are at most four conjugacy classes of involution which are not in C and whose centralizers have rank 2. Hence there are at most four conjugacy classes of maximal elementary abelian subgroups of rank 2 that are not in C. This completes the proof. □

Proof of Theorem 1.1. We noted in the introduction that Alperin [1] has shown that any two elementary abelian subgroups of rank 3 in G are in the same connected component of P/G. Specifically, what he proves is that if G is a p-group and if E and F are elementary abelian subgroups of rank 3 in G, then there is a chain of subgroups E = E_{0}, E_{1}, …, E_{n} = F such that E_{i−1} ∩ E_{i} has rank 2 for all i = 1, …, n. Therefore the components of P/G consist of the component that contains all elementary abelian subgroups of rank 3 or more (if there is one) and the isolated points consisting of the conjugacy classes of maximal elementary abelian subgroups of rank 2. The theorem now follows from Theorem 1.3. □

4. Some remarks

It was said in the introduction that the main theorems were predicted with some experimental evidence. We remark here that the upper bound on the number of components is achieved in a reasonably small group. Specifically, the extraspecial group of order 32, which is a central product of a dihedral group of order 8 and a quaternion group of order 4, has five maximal elementary abelian subgroup of rank 2, all of which are normal. This is Magma small group number 50. A check using the computer algebra system MAGMA [4] revealed no higher number of components in 2-groups up to order 256.

It seems likely that results similar to Theorems 1.1 and 1.3 should be true in the case that p is an odd prime. Again this is confirmed by a small amount of experimental evidence. In this case it appears that the upper bound might be p + 1 on the number of components of the orbit space P/G of the poset of elementary abelian subgroups of rank at least 2. Alperin [2] has recently shown that there is an upper bound on the number of components, though the actual value of the upper bound is elusive.

Subsequent to the announcement of these results, Alperin [2] has also provided some specifics on the structure of the subgroup of a 2-group generated by noncentral involutions whose centralizers have rank 2.

References

[1] J. Alperin, A construction of endo-permutation modules, J. Group Theory 4(2001), 3–10.

[2] J. Alperin, Private Communication.

[3] D. J. Benson, “Representations and Cohomology I, II”, Cambridge Univ. Press, Cambridge, 1991.

[4] W. Bosma and J. Cannon, Handbook of Magma Functions, Magma Computer Algebra, Sydney, 2002.

[5] S. Bouc and J. Thévanaz, Gluing torsion endo-permutation modules, (in preparation).

[6] J. Carlson, Constructing endotrivial modules, J. Pure and Appl. Algebra, (to appear).

[7] J. Carlson, N. Mazza and D. Nakano Endotrivial modules for finite groups of Lie type, J. Reine Angew. Math. (to appear).

[8] J. Carlson, L. Townsley, L. Valero-Elizondo and M. Zhang, Cohomology Rings of Finite Groups, Kluwer, Dordrecht, 2003.

[9] Z. Janko Finite 2-groups with no normal elementary abelian subgroups of order 8, J. Algebra 246(2001), 951-961.

Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA

  E-mail address: jfc@@math.uga.edu