Set Covering and Serre's Theorem on the Cohomology Algebra of a
p-Group
Erg"un Yalcin
June 23, 2000
Abstract
We define a group theoretical invariant, denoted by s(G), as a solution *
*of a certain set covering problem,
and show that it is closely related to chl(G), the cohomology length of a p*
*-group G. By studying s(G), we
improve the known upper bounds for the cohomology length of a p-group, and *
*determine chl(G) completely
for extra-special 2-groups of real type.
1991 Mathematics Subject Classification. Primary: 20J06; Secondary: 20D15, *
*20D60, 51E20.
Key words and phrases. Cohomology length, extra-special p-group, set coveri*
*ng problem, ovoid.
1 Introduction
A classical theorem by Serre [13] states that if G is a p-group which is not el*
*ementary abelian
then there exist non-zero elements u1; u2; : :;:um 2 H1(G; Fp) such that
u1u2. .u.m= 0 if p = 2;
fi(u1)fi(u2) . .f.i(um )i=f0 p >:2
Since its first appearance many different proofs have been given for this theor*
*em, and there
has been some interest in finding the minimum number of one dimensional classes*
* required for
a vanishing product. For a p-group G, which is not elementary abelian, the smal*
*lest integer m
that satisfies the conclusion of Serre's Theorem is usually referred as the coh*
*omology length of
G and denoted by chl(G). Since we will be studying this invariant, throughout t*
*he paper we
assume that the groups considered are not elementary abelian unless otherwise s*
*pecified.
Although Serre's original proof does not provide an estimate for chl(G), lat*
*er proofs, in-
cluding an independent proof by Serre himself, give upper bounds for chl(G) in *
*terms of
k = dimFpH1(G; Fp) and the prime p. The first known upper bound, chl(G) pk - 1*
*, was
k-1
given by O. Kroll [6]. This was improved by Serre [14] to chl(G) p___p-1, an*
*d later by
T. Okuyama and H. Sasaki [9] to chl(G) (p + 1)pk-2. The most recent result sta*
*tes that
k_]-1
chl(G) (p + 1)p[2 , which is due to P.A. Minh [7].
1
In his proof of Serre's Theorem, P.A. Minh uses a detection theorem for extr*
*a-special p-
groups. We will exploit this theorem further to relate the cohomology length t*
*o a group
theoretical and combinatorial invariant defined as follows: Let G be a p-group *
*and S be a
subset of G. We say S is a representing set if it includes at least one non-cen*
*tral element from
each maximal elementary abelian subgroup of G. If G is not p-central, i.e., if*
* there exists
an element of order p which is not central, then one can always find a represen*
*ting set for
G. For a p-group G which is not p-central we define s(G) as the minimum cardina*
*lity of a
representing set for G. To make our statements more general, we are using Quill*
*en's definition
of extra-special p-groups which only requires Frattini subgroup to be a cyclic *
*group of order
p (see Section 2 for properties of such groups.) We prove the following:
Proposition 1.1. If G is an extra-special p-group which is not p-central, then *
*chl(G) s(G).
Moreover, if G has self-centralizing maximal elementary abelian subgroups, the*
*n chl(G) =
s(G).
As a combinatorial problem computing s(G) is a set covering problem (See [3]*
*) where the
adjacency matrix has a very special form due to the special structure of extra-*
*special p-groups.
Although there exist algorithms to solve set covering problems, and they provid*
*e upper bounds
for minimal order covering set, these results are too general and do not provid*
*e sufficiently
sharp bounds in our case. Using some counting arguments for extra-special p-gro*
*ups with a
small number of generators, we find an upper bound for s(G), and conclude:
Theorem 1.2. If G is a p-group and k = dimFpH1(G; Fp), then chl(G) p + 1 if k*
* 3 and
k_]-2
chl(G) (p2 + p - 1)p[2 if k 4:
In Section 5, we calculate the invariant s(G) when G is an extra-special 2-g*
*roup of real
type, i.e., G is isomorphic to central product of D8's. We conclude:
Theorem 1.3. Let Gn be an extra-special 2-group isomorphic to an n-fold centra*
*l product of
D8's. Then,
(
2n-1 + 1 if n 4;
chl(Gn) =
2n-1 + 2n-4 if n 5:
For an extra-special 2-group G, the invariant s(G) is equal to the minimum c*
*ardinality of a
set of points that meets every maximal subspace of the associated quadric. Such*
* sets are called
sets closest to ovoids and was first introduced by K. Metsch [8]. By Metsch's c*
*alculations for
elliptic quadrics, we determine s(G) for the corresponding family of extra-spec*
*ial 2-groups.
We also do similar calculations for parabolic quadrics to complete the calculat*
*ions of s(G) for
extra-special 2-groups.
We conclude the paper with a discussion of open problems.
2
Throughout the paper we use the following notations: For a group G, Z(G), (G*
*) and
[G; G] denote the center, the Frattini subgroup and the commutator subgroup of *
*G respec-
tively. We write C G(g) for the centralizer of an element g 2 G, and C S(g) fo*
*r the set of
elements in a set S G that commutes with g. As it was mentioned before, we use*
* Quillen's
definition of extra-special p-groups since this is the form we have most of the*
* cohomological
information. Notice that this is different than usually accepted group theoreti*
*cal definition of
extra-special p-group where one must have (G) = [G; G] = Z(G). Finally, all the*
* cohomology
groups in the paper are in Z=p coefficients and they are simply denoted by H*(G*
*).
Acknowledgments: I would like to thank Jonathan Pakianathan for many helpful*
* conver-
sations and especially for pointing out the references on ovoids. I am also gre*
*atful to Jon F.
Carlson for his comments on the earlier version of this paper, and would like t*
*o acknowledge
that some of the results in Section 3 was already known to him.
2 Properties of Extra-special p-Groups
Let G be an extra-special p-group. By this we mean that G fits into an extension
1 ! Z=p ! G ! V ! 1
where V is a vector space over Z=p. If G ~=G0xZ=p for some G0 G, then chl(G) =*
* chl(G0)
and s(G) = s(G0). Hence, without loss of generality we can assume that G has n*
*o proper
direct factors. It follows that if G is represented by the extension class ff 2*
* H1(V ), then there
exists a basis such that ff is in one of the following forms (See [7]):
-for p = 2 and k = 2n: (a) x1y1 + x2y2 + . .+.xnyn
(b) x21+ y21+ x1y1 + x2y2 + . .+.xnyn
-for p = 2 and k = 2n + 1:(c) x20+ x1y1 + x2y2 + . .+.xnyn
-for p > 2 and k = 2n: (d) x1y1 + x2y2 + . .+.xnyn
(e) fi(x1) + x1y1 + x2y2 + . .+.xnyn
-for p > 2 and k = 2n + 1:(f) fi(x0) + x1y1 + x2y2 + . .+.xnyn
where k = dimZpV . From this it is easy to see that every maximal elementary ab*
*elian subgroup
of G is of rank n when ff is in the form (b) and n+1 otherwise. Observe also th*
*at if G is of the
form (a) or (d), then C G(E) = E for every maximal elementary abelian subgroup *
*E G. It
is well known that when G is an extra-special p-group of type (e) or (f), then *
*chl(G) p (see
page 107 in [1].) Therefore, we can ignore these cases and assume that all the *
*extra-special
p-groups are in one of the forms (a) - (d).
Let {a1; : :;:an; b1; : :;:bn} be a set of generators of G satisfying xi(ak)*
* = ffii;kand yi(bk) =
ffii;kwith ffi being the Kronecker symbol (when k is odd, we take {a0; a1; : :;*
*:an; b1; : :;:bn} as
a set of generators.) Let Gm denote the subgroup generated by {ai; bi | i m}, *
*then Gm is
3
isomorphic to the central product G
m-1 * . In general, when G is an *
*extra-special
group with k = 2n or 2n + 1, we write G = Gn for convenience.
In the following sections we use some counting arguments involving the numbe*
*r of maximal
elementary abelian subgroups of Gn. We do this calculation here (See also [12] *
*or [5]):
Lemma 2.1. Let t(Gn) denote the number of maximal elementary abelian subgrou*
*ps of Gn.
Then for each of the cases listed above, we have
Qn Qn
(a) t(Gn) = (2i-1+ 1), (b) t(G1) = 1; t(Gn) = (2i+ 1) for n 2,
i=1n i=2
Q Qn
(c) t(Gn) = (2i+ 1), (d) t(Gn) = (pi+ 1).
i=1 i=1
Proof.It is easy to verify the value of t(G1) for each case, so we can assume n*
* 2. Consider
the set of pairs (E; g) where E is a maximal elementary abelian subgroup of Gn *
*and g is an
element in E which is not central. It is easy to see that the order of this set*
* is t(Gn).(|E|-p).
On the other hand, we can count this set starting from non-central elements of *
*order p.
Observe that if g is an element of order p that is not central, then the centra*
*lizer C G(g) is
isomorphic to Gn-1x Z=p where Gn-1 Gn is the subgroup generated by {ai; bi| i *
* n - 1}.
So, g is included in exactly t(Gn-1) maximal elementary abelian subgroups. If (*
*Gn) is the
number of elements in Gn of order less than equal to p, then the order of above*
* set is equal
to ((Gn) - p) . t(Gn-1). Setting the results of two different ways of counting *
*equal, we find
that t(Gn)=t(Gn-1) = ((Gn) - p)=(|E| - p).
In the case of (d), (Gn) = p2n+1and |E| = pn+1. Hence,
_t(Gn)_ p2n+1- p n
= _________= p + 1;
t(Gn-1) pn+1 - p
therefore t(Gn) = (pn + 1)(pn-1 + 1) . .(.p + 1). For p = 2, we need to calcula*
*te (Gn) since
its value is not obvious. We do this by using a recursive relation:
Observe that Gn is covered by anGn-1, eGn-1, bnGn-1 and anbnGn-1 which are t*
*he cosets
of the subgroup Gn-1. Therefore we have
(Gn) = 3(Gn-1) + (|Gn-1| - (Gn-1))
from which we obtain
(Gn) = 2n-1(G1) + 2n-2(2n-1 - 1)|G1|
for all n 2. Substituting the values of (G1) and |G1| for each case, we find t*
*hat (Gn) =
22n+ 2n in the case of (a), (Gn) = 22n- 2n for n 2 in the case of (b), and (Gn*
*) = 22n+1
*
* __
in the case of (c). Using these we obtain the values of t(Gn) as listed in the *
*lemma. |__|
4
Lemma 2.2. Let E
r Gn be an elementary abelian subgroup of rank r < rk(Gn). T*
*hen, the
number of maximal elementary abelian subgroups in Gn that include Er is equal t*
*o t(Gn-r+1),
where Gn-r+1 Gn is the subgroup generated by {ai; bi | i n - r + 1}.
Proof.If E is a maximal elementary abelian subgroup that includes Er, then E C*
*G (Er) ~=
(Er=(G)) x Gn-r+1. Hence, maximal elementary abelian subgroups that include Er *
*are in
*
* __
one to one correspondence with maximal elementary abelian subgroups of Gn-r+1. *
* |__|
3 Representing Sets
In this section we discuss the notion of representing sets and prove Propositio*
*n 1.1.
Definition 3.1.Let G be a p-group, and S be a subset of G. We say S is a repres*
*enting set
for G if S \ (E - Z(G)) 6= ; for every maximal elementary abelian subgroup E *
*G.
If G is a p-group which is not p-central, i.e. it has an element of order p*
* which is not
central, one can always find a representing set for G. For a p-group G which is*
* not p-central
we define s(G) as the minimum of cardinalities of representing sets for G. Thi*
*s restriction
does not affect our results, since we will be working with extra-special p-grou*
*ps and the only
p-central p-group in this class is the group of unit quaternions Q8 which is kn*
*own to have
cohomology length 3. We also would like to note that when p is odd, p central g*
*roups have
rather small cohomology length:
Proposition 3.2. If p is an odd prime and G is a p-central p-group, then chl(G)*
* p.
Proof.We will be using two well known facts about the cohomology length of a p-*
*group (for
further information on Serre's theorem see [1], [2], or [4]). If L is a factor *
*group of a p-group
G, i.e. L ~=G=N for some normal subgroup N G, then chl(G) chl(L). This is a d*
*irect
consequence of the fact that if ss is the quotient map G ! L = G=K, then the in*
*duced map
ss* : H1(L) ! H1(G) is injective. Also recall that the cohomology length of an *
*extra-special
p-group of type (e) or type (f) is known to be less than p. Therefore, to prove*
* the result we
only need to show that a p-central p-group (p is odd) which is not elementary a*
*belian always
has a factor group isomorphic to an extra-special p-group of exponent p2.
Let G be a p-central p-group where p is an odd prime. Recall that a p-centr*
*al group is
a group where every element of order p is central. So, it has a unique maximal *
*elementary
abelian subgroup. Moreover, when p is odd, taking the quotient with the maximal*
* elementary
abelian subgroup gives again a p-central p-group. So, If E is the maximal eleme*
*ntary abelian
subgroup of G and if the quotient group G=E is not elementary abelian, G will h*
*ave the desired
factor group by induction. Thus we can assume G=E is elementary abelian. Furthe*
*rmore, we
5
can assume that the Frattini subgroup (G) is equal to E, because otherwise G ~
= *
*G0 x Z=p
for some p-central subgroup G0 G, and again the result follows by induction.
Now, since the group G itself is not elementary abelian, the subgroup of Gp *
* G generated
by the pth powers is nontrivial and it is included in (G) = E. Let M be a maxi*
*mal
subgroup of E that does not include Gp. Then, the factor group G=M is an extra*
*-special
*
* __
p-group (possibly abelian) of exponent p2. So, the proof is complete. *
* |__|
Remark 3.3. The conclusion of Proposition 3.2 is not true for 2-central groups*
*. The simplest
example is the case G = Q8 where the group is 2-central but the cohomology leng*
*th is equal
to 3.
We now quote an important detection theorem for the cohomology of extra-spec*
*ial p-groups.
Let EA G be the set of all maximal elementary abelian subgroups of G, and let H*
*(G) denote
the subalgebra of H*(G) generated by one dimensional classes when p = 2 and Boc*
*ksteins of
one dimensional classes when p > 2.
Theorem 3.4 (Quillen [12 ], Tezuka-Yagita [15 ]).The map
Y Y
resGE: H(G) ! H*(C G(E))
E2EAG E2EAG
is an injection.
Now, we introduce some notation: Given an element s 2 G - Z(G), we can defi*
*ne an
homomorphism s : G ! Z=p by letting (g) = [s; g] for all g 2 G. Since H1(G) ~=
Hom (G; Z=p), the homomorphism uniquely defines a one dimensional cohomology c*
*lass
which we will denote by us. Note that us satisfies the property CG (s) = kerus.*
* For simplicity,
let vs denote fi(us) when p is odd and us when p = 2.
Lemma 3.5.Q If S is a representing set for an extra-special p-group G which is*
* not p-central,
then oe = vs = 0 in H*(G). Hence, chl(G) s(G).
s2S
Proof.Let S be a representing set for G. For every E 2 EA G, there exists a no*
*n-central
element s 2 S such that s 2 E, i.e., C G(E) CG (s) = kerus. Therefore,
resGCG(E)vs = resCG(s)CG(E)resGCG(s)vs = 0:
*
* __
Thus resGCG(E)oe = 0 for every E 2 EA G. Applying Theorem 3.4, we obtain oe = 0*
*. |__|
Remark 3.6. In fact, Lemma 3.5 holds for a larger class of groups. Let G be a *
*p-group which
is not p-central. Suppose that G satisfies the above detection theorem, for exa*
*mple, G is such
that H*(G) is Cohen-Macaulay. If S is a representing set for G, then for every *
*s 2 S we can
find a one dimensionalQclass us such that C G(s) kerus. As in the proof of Lem*
*ma 3.5, we
can conclude vs = 0. Hence, chl(G) s(G).
s2S
6
If G is an extra-special p-group of type (a) or (d), we will prove conversel*
*y that given a
vanishing product of m one dimensional classes (respectively vanishing product *
*of Bocksteins
of m one dimensional classes when p is odd), one can find a representing set of*
* order less than
m. For this we first observe that if G is an extra-special p-group of type (a) *
*or (d), and M is
a index p subgroup of G, then |Z(M)| > |Z(G)|, i.e. there is an element g 2 G -*
* Z(G) such
that M = CG (g). So, for every u 2 H1(G; Fp) there is a non-central element g 2*
* G such that
CG(g) = keru. As above, let videnote fi(ui) when p is odd and uiwhen p = 2
Lemma 3.7. Let G be an extra-special p-group of the form (a) or (d). Let {u1*
*; u2; : :;:um }
be a set of classes in H1(G), and S = {g1; :::; gm } be a set of corresponding *
*group elements
mQ
which satisfy C G(gi) = kerui for i = 1; : :;:m. If oe = vi= 0, then S is a *
*representing set
i=1
for G. Hence, s(G) chl(G).
Proof.Take an element E 2 EA G. Note that the images of viunder the restriction*
* map resGE:
mQ
H*(G) ! H*(E) lie in the polinomial subalgebra of H*(E). Thus, resGEoe = res*
*GEvi = 0
i=1
implies that resGEvi = 0 for some i. It follows that E ker ui = C G(gi), and *
*hence,
gi 2 C G(E) = E. Since this is true for all E 2 EA G, we conclude that S is a r*
*epresenting
*
* __
set. *
* |__|
*
* __
Proof of Proposition 1.1.Follows from Lemma 3.5 and Lemma 3.7. *
* |__|
Remark 3.8. Another obvious choice for a definition of representing set is the*
* one that
requires the set to include at least one non-central element from each maximal *
*abelian sub-
group. For any non-abelian p-group G, we can always find a set that represents*
* maximal
abelian subgroups, so we can define sA(G) as the order of smallest such set in *
*G. Using
Minh's arguments in [7], one can show that chl(Gn) sA(Gn). However, in the ca*
*se of a
extra-special 2-group of type (a), this gives a weaker upper bound. For example*
* in the case
of G = D8, chl(G) = s(G) = 2 but sA(G) = 3. In general the following relations *
*are known
for each type:
(a) chl(G) = s(G) < sA(G),
(b) chl(G) sA(G) s(G),
(c) chl(G) s(G) = sA(G).
We conclude this section with some examples of representing sets from which *
*we obtain
some of the previously known upper bounds for chl(G). In all the examples below*
*, G is an
extra-special p-group which is not p-central, and k = dimFpH1(G).
Example 3.9. For every nontrivial cyclic subgroup C G=(G), choose an element *
*g 2 G
such that the image of g under quotient map generates C. Let S be the set of a*
*ll chosen
7
elements. It is easy to see that every maximal elementary abelian subgroup incl*
*udes a sub-
group of the form <(G); g> for some noncentral element g 2 G. Hence S is a repr*
*esenting
set. Since the set S is in one to one correspondence with the projective space *
*of the vector
k-1
space V = G=(G), we obtain Serre's upper bound: chl(G) |S| |P (V )| = p___p-1.
Example 3.10. Let H G be an index p2 subgroup of G such that rk(H) = rk(G) - *
*1. Let
S be as in the previous example, and S0 = S \ (G - H). Since every maximal ele*
*mentary
abelian subgroup includes at least one element from G - H, it includes a subgro*
*up of the
form <(G); g> for some g 2 G - H. Hence S0is a representing set. This gives Oku*
*yama and
Sasaki's bound: chl(G) |S0| = (pk+1- pk-1)=(p2 - p) = (p + 1)pk-2.
Example 3.11. Minh's upper bound was obtained using maximal abelian subgroups.*
* So, we
will construct a set that represents maximal abelian subgroups. Let A be a maxi*
*mal abelian
subgroup in Gn and A0 be a index p subgroup of A that includes the center. We *
*form S
by picking one non-central element from each abelian subgroup of the form where
Z(G) is the center of Gn and g is an element in C G(A0) - A0. It is easy to se*
*e that every
maximal abelian subgroup has a nonempty intersection with C G(A0) - A0, so S is*
* a set that
represents maximal elementary abelian subgroups. This gives Minh's upper bound:
k_]-1
chl(G) |S| (p2 - 1)|A0|=(p - 1)|Z(G)| = (p + 1)p[2 :
4 Proof of Theorem 1.2
In this section we prove Theorem 1.2 stated in the introduction. We continue t*
*o use the
notation introduced in Section 2. In particular, Gn denotes an extra-special g*
*roup with
k = 2n or k = 2n + 1 with a basis {ai; bi | i n} chosen as in Section 2. For e*
*very m n,
Gm Gn denotes the subgroup generated by {ai; bi| i m}.
Lemma 4.1. s(G2) p2 + p - 1.
Proof.For each case listed in Section 2, we show that there is a representing s*
*et of order less
than p2 + p - 1 by using the calculations done for t(G2) in Lemma 2.1.
(a) Let S = {a1; b1; a1b1a2b2}. Every element in S is included in t(G1) = 2 max*
*imal elementary
abelian subgroups. Since elements in S are pairwise non-commuting, there are no*
* maximal
elementary abelian subgroups that include two elements in S. So, elements in S*
* represent
6 distinct maximal elementary abelian subgroups. Since t(G2) = 6, we conclude t*
*hat S is a
representing set.
(b) In this case there are only 5 maximal elementary abelian subgroups which ar*
*e , ,
, , and . Hence, the set S = {a2; b2; a1a2b2*
*; a1b1a2b2; b1a2b2} is
a representing set.
8
(c) Let S = {a
1; b1; a0a1b1a2; a0a1b1b2; a1b1a2b2}. Each element s 2 S represe*
*nts t(G1) = 3
maximal elementary abelian subgroups. The elements in S are choosen in such a *
*way that
they are pairwise non-commuting. Therefore, none of the two appears in the same*
* maximal
elementary abelian subgroup, thus S represents the total of 15 distinct maximal*
* elementary
abelian subgroups. But, this is all there is since t(G2) = (22 + 1)(2 + 1) = 1*
*5. So, S is a
representing set.
(d) For each nontrivial element in G1=(G1) we choose a representative in G1 and*
* form
the set S1. Now let S2 = {ai2b2 | i = 0; 1; : :;:(p - 1)}, and S = a2S1 [ S2. *
* It is clear
that |S| = |S1| + |S2| = p2 + p - 1. Let E be a maximal elementary abelian sub*
*group
of G2. If E \ S2 = ;, then there is an element g 2 E that commutes with none o*
*f the
elements in S2. Since the centralizers of elements in {a2} [ S2 cover G, the e*
*lement g lies
in C G(a2) - G1 = a2G1. Therefore \ S1 6= ;. Hence S is a representi*
*ng set. (It
is possible to reach the same conclusion through a counting argument. S1 divide*
*s into p + 1
subsets where each subset represents [1 + p(p - 1)] maximal elementary abelian *
*subgroups,
and each element in S2 represents p + 1 maximal elementary abelian subgroups. S*
*o, we get a
total of (p + 1)[1 + p(p - 1)] + p(p + 1) = (p2+ 1)(p + 1) distinct maximal ele*
*mentary abelian
*
* __
subgroups represented which are all there is in G2.) *
* |__|
Lemma 4.2. s(Gn) p . s(Gn-1) for n 3.
Proof.Let S0 be a representing set for Gn-1 with |S0| = s(Gn-1). Set S = {ains|*
*s 2 S0; i =
0; 1; : :;:(p - 1)}. Let E be a maximal elementary abelian subgroup of Gn. If E*
* \ Gn-1 is a
maximal elementary abelian subgroup of Gn-1, then E is represented by S0 and he*
*nce by S.
Otherwise E = for some x; y 2 Gn-1 where E0 Gn-1 is an elementar*
*y abelian
subgroup of rank n - 1. Observe that is a maximal elementary abelian su*
*bgroup of
Gn-1, so there is an s 2 S0 which represents . Then, ains 2 E for some *
*i and hence
*
* __
S \ (E - Z(G)) 6= ;. Thus, S is a representing set for Gn and s(Gn) |S| = p . *
*s(Gn-1). |__|
Now, we prove our main theorem.
Proof of Theorem 1.2.By Lemma 4.1 and Lemma 4.2, we obtain s(Gm ) (p2 + p - 1)*
*pm-2
for m 2. By Proposition 1.1, it follows that chl(Gm ) (p2 + p - 1)pm-2 when m*
* 2. By
earlier calculations we also know that chl(G1) p + 1. We can extend this resul*
*t to arbitrary
p-groups as follows: Let G be a p-group with k = H1(G) = 2n or 2n + 1. Then, G*
* has a
factor group isomorphic to some Gm with m n. For any factor group L of G, we *
*have
*
* __
chl(G) chl(L), so the theorem follows. *
* |__|
9
5 Calculations for extra-special 2-groups of type (a)
In this section we compute the invariant s(G) for extra-special 2-groups of typ*
*e (a) and
obtain Theorem 1.3 as a corollary. Let Gn denote an extra-special 2-group of ty*
*pe (a) with
dimF2H1(G) = 2n. The number of maximal elementary abelian subgroups in Gn is ca*
*lculated
in Section 2 as n
Y
t(Gn) = (2i-1+ 1)
i=1
and by Lemma 2.2, every noncentral element of order 2 is included in exactly t(*
*Gn-1) maximal
elementary abelian subgroups. So, if S is a representing set, then it must hav*
*e at least
t(Gn)=t(Gn-1) = 2n-1 + 1 elements. Thus,
Lemma 5.1. s(Gn) 2n-1 + 1.
Observe that a set of noncentral elements of order 2 with |S| = 2n-1 + 1 is *
*a representing
set if and only if every maximal elementary abelian subgroup includes only one *
*element from
S. The last statement is true if and only if the elements in S are pairwise no*
*n-commuting
(such a set is called a non-commuting set.) We conclude:
Lemma 5.2. Let S be a set of elements of order 2 in Gn such that |S| = 2n-1 +*
* 1. Then, S
is a representing set if and only if S is a non-commuting set.
Now, the following is a easy consequence of these Lemmas:
Lemma 5.3. s(Gn) = 2n-1 + 1 for n 4.
Proof.Let {a1; b1; : :;:an; bn} be a generating set as described in Section 2. *
* By previous
lemmas, it is enough to find a subset Sn Gn of order 2n-1 + 1 such that Sn is *
*a set of
pairwise non-commuting elements of order 2. Let
S1 = {a1; b1}; S2 = {a1; b1; a1b1a2b2}; S3 = {a1; b1; a1b1a2a3b3; a1b1b2*
*a3b3; a1b1a2b2};
S4 = {a1; b1; a1b1a2a4b4; a1b1b2a4b4; a1b1a2b2a3; a1b1a2b2b3; a1b1a3b3a4; a1b*
*1a3b3b4;
a1b1a2b2a3b3a4b*
*4}:
It is straightforward to check that elements in each Si are of order 2 and pair*
*wise non-
*
* __
commuting. *
* |__|
Unfortunately, the equality s(Gn) = 2n-1 + 1 does not hold in general. This *
*is because in
general the orders of non-commuting sets in Gn are much smaller than 2n-1 + 1. *
*Let nc(G)
denote the order of largest non-commuting set in G. The following calculation w*
*as originally
done by Marty Isaacs (See [11]):
10
2n+1
Lemma 5.4. If G is an extra-special 2-group of order 2 , then nc(G) = 2n +*
* 1.
Proof.It is easy to see that there is a non-commuting set of order 2n + 1 defin*
*ed inductively
by letting Xn = {an} [ {bn} [ anbnXn-1 and X1 = {a1; b1; a1b1}. Now we will sho*
*w that one
can not find a non-commuting set larger than this. Let X be a set of pairwise n*
*on-commuting
elements in Gn. Take two elements x and y in X. Observe that the rest of the el*
*ements in X
should lie in the coset xyCG(). Since CG() is an extra-special grou*
*p of order 22n-1,
*
* __
by induction |X| - 2 2n - 1 and hence |X| 2n + 1. We conclude that nc(G) = 2n*
* + 1. |__|
Lemma 5.5. 2n-1 + 1 < s(Gn) 2n-1 + 2n-4 for every n 5.
Proof.When n 5, we have 2n + 1 < 2n-1+ 1. So, by above lemmas, 2n-1+ 1 < s(Gn)*
*. The
*
* __
second inequality follows from Lemma 5.3 and Lemma 4.2. *
* |__|
As stated in Theorem 1.3, we claim that s(Gn) = 2n-1 + 2n-4 when n 5. For t*
*he proof
we need the following:
Lemma 5.6. Let S be a representing set for Gn with minimum order. Then, for e*
*very s 2 S,
there exist at least 2n-1 elements in S that do not commute with s, i.e.,
|S - CS(s)| 2n-1 for everys 2 S:
Proof.Take an element s 2 S. Observe that there exists a maximal elementary ab*
*elian
subgroup E Gn such that E \ S = {s}. Because, otherwise S - {s} is a represent*
*ing set
with smaller order, contradicting the assumption that S is a representing set w*
*ith minimum
order. Consider the set {E01; : :;:E0m} of index 2 subgroups of E that include*
* the center of
Gn, but do not include the element s. An easy calculation shows that there are*
* exactly
(2n - 1) - (2n-1- 1) = 2n-1 such subgroups, so m = 2n-1. Each E0iis included in*
* 2 maximal
elementary abelian subgroups one of which is E. For each E0iwe call the other *
*maximal
elementary abelian subgroup Ei. Since Ei\ S 6= ;, for each i there is an elemen*
*t si2 S such
that si2 Ei- E0i. If si= sj for some i 6= j, then sicommutes with both E0iand E*
*0j, and hence
commutes with EiEj = E. Then, si2 E \ Ei= E0iwhich is in contradiction with E0i*
*\ S = ;.
So, si's are distinct. Finally, if for some i the element sicommutes with s, th*
*en sicommutes
with = E, and hence si2 E \ Ei= E0i. This again leads to a contadiction*
*, so si's do
*
* __
not commute with s 2 S. Hence the proof of the lemma is complete. *
* |__|
For the proof of claim s(Gn) = 2n-1 + 2n-4 for n 5, it remains to show that*
* if S is a
representing set, then there exists an element s 2 S such that s commutes with *
*at least 2n-4
elements in S, i.e. |C S(s)| 2n-4. For this we need a stronger version of the *
*above lemma:
Lemma 5.7. Let S be a representing set for Gn with minimum order and let g be*
* a non-
central element G which is not included in S. Suppose further that there exist*
*s a maximal
11
elementary abelian subgroup E G
n such that g 2 E and S \ E = {s}. Then, there *
*exist at
least 2n-2 elements in S that do not commute with g.
Proof.The proof is similar to the proof of Lemma 5.6. In this case we take the *
*set {E01; : :;:E0m}
as the set of index 2 subgroups of E that include the center and the element gs*
*, but do not
include s. Counting such subgroups we find that m = 2n-2. Observe that the el*
*ements
s1; : :;:sm , chosen as in the proof of Lemma 5.6, will commute with gs, but th*
*ey will not
*
* __
commute with s, hence will not commute with g. *
* |__|
Lemma 5.8. If S is a representing set for Gn, then there is an element s 2 S *
*such that
|C S(s)| 2n-4:
Proof.The lemma is true for n 4, so assume n 5. Since 2n + 1 < 2n-1 + 1 < |S*
*| for
n 5, there exist a; b 2 S such that [a; b] = 1. Without loss of generality we *
*can assume S
is a representing set with minimum order. Then by Lemma 5.6, we have |S - CS(a)*
*| 2n-1.
If |C S(b)| 2n-4, then we are done. So, assume |C S(b)| < 2n-4. Since the co*
*mmutator
subgroup of Gn is isomorphic to Z=2, every element in G commutes with a, b or a*
*b. Therefore
C S(a) [ CS(b) [ CS(ab) = S. Hence |C S(ab)| > 2n-1 - 2n-4 > 2n-4. If ab 2 S th*
*en we are
done. So, assume ab 62 S.
If there exists a maximal elementary abelian subgroup E Gn such that ab 2 E*
* and |E \
S|=1, then by Lemma 5.7, we have |S - CS(ab)| 2n-2, which implies either |C S(*
*a)| 2n-3
or |C S(b)| 2n-3, hence the lemma will be true. So, assume contrary that for e*
*very maximal
elementary abelian subgroup E G that includes the element ab, we have |S \ E| *
* 2. Now
we consider the following cases:
Case 1: Suppose that there exists an element s 2 S such that abs 62 S. Let *
*E3 be the
subgroup generated by the elements ab and s, and the central element c 2 Gn. Le*
*t E3 denote
the set of maximal elementary abelian subgroups in Gn that include E3. By minim*
*ality of S,
we have S \ cS = ;, and by above assumption abs 62 S. So, E3\ S = {s}. By Lemma*
* 2.2, E3
is included in t(Gn-2) maximal elementary abelian subgroups, i.e. |E3| = t(Gn-2*
*). Recall that
for every maximal elementary abelian subgroup E, we have |E \ S| 2. So, for ev*
*ery E 2 E3,
we have E \ (S - {s}) 6= ;. Note that an element s02 S - {s} is included in t(G*
*n-3) maximal
elementary abelian subgroups in E3. So, there are at least t(Gn-2)=t(Gn-3) = 2n*
*-3+1 elements
in S - {s} which are included in a maximal elementary abelian subgroup E 2 E3. *
*Since each
of these elements will commute with s, we conclude that |C S(s)| 2n-3 + 2 2n-*
*4.
Case 2: Suppose that for every s 2 C S(ab), abs 2 S, and there exists an ele*
*ment x 2 S
such that [x; ab] 6= 1. Then, x commutes with either s or abs for every s 2 S. *
*Since C S(ab)
can be written as disjoint union of X and abX for some X S, we obtain
|C S(x)| |C S(ab)|=2 > 2n-2 - 2n-5 > 2n-4:
12
Case 3: Finally, we assume that for every s 2 C
S(ab), abs 2 S, and S C S(a*
*b). Take
an element x 2 Gn of order 2 such that [ab; x] 6= 1. Let H Gn be the centrali*
*zer of
. Subgroup H is isomorphic to Gn-1 and C G(ab) is a disjoint union of H *
*and abH. So,
S CG (ab) can be written as a disjoint union (S \ H) t (S \ abH). Since S = ab*
*S, we have
S \ H = abS \ H, and hence ab(S \ H) = S \ abH. So, S = S0t abS0where S0= S \ H*
* H.
Now, we claim that S0 H is a representing set for H. Let E0 be a maximal ele*
*mentary
abelian subgroup of H. Then, E = is a maximal elementary abelian subgr*
*oup for Gn,
and hence E \ S 6= ;. Since E = E0t abE0, we have
E \ S = (E0t abE0) \ (S0t abS0) = (E0\ S0) t ab(E0\ S0):
Thus E0\ S06= ;. Therefore, S0is a representing set for H. By induction there i*
*s an element
s02 S0such that |C S0(s0)| 2n-5. Since |C S(s0)| = 2 |C S0(s0)|, we obtain tha*
*t |C S(s0)| 2n-4
for s02 S.
*
* __
Since we have considered all the possible cases, the proof of the lemma is c*
*omplete. |__|
Proof of Theorem 1.3.By Lemma 1.1, we have s(Gn) = chl(Gn), so it is enough to *
*prove the
Theorem for s(Gn). By Lemma 5.3, we know s(Gn) = 2n-1 + 1 for all n 4, and by *
*Lemma
5.5, we have 2n-1 + 1 < s(Gn) 2n-1 + 2n-4 for all n 5. Finally, Lemma 5.6 and*
* Lemma
*
* __
5.8 implies that s(Gn) 2n-1 + 2n-4. So, the proof is complete. *
* |__|
6 Calculations for other types of extra-special 2-groups
The arguments used in the previous section can easily be extended to other type*
*s of extra-
special 2-groups to calculate s(G) for these groups. In fact s(G) corresponds t*
*o an invariant
in combinatorial algebra and the case of type (b) has already been calculated b*
*y Klaus Metsch
[8]. We explain here beliefly the relation between representing sets and the s*
*ets which are
called sets closest to ovoids.
Let P G(n; q) denote the projective space of (n + 1)-dimensional vector spac*
*e over F q,
the finite field of q-elements. A non-singular quadric in P G(n; q) is the var*
*iety V (F ) of a
non-singular quadratic form
Xn X
F = aix2i+ aijxixj:
i=0 i 2n + 1
for n 3. By Lemma 4.2, we also know s(Gn) 2n + 2n-2. Now, let S be a represen*
*ting
set of minimal order, we will show that |S| 2n + 2n-2. This will complete the *
*proof of the
proposition.
The argument in Lemma 5.6 can be repeated easily for this case. Since index *
*2 subgroups of
a maximal elementary abelian groups are included in 3 maximal elementary abelia*
*n subgroups,
we find |S -C S(s)| 2n for every s 2 S. Now, we will show inductively that the*
*re is an s 2 S
such that |C S(s)| 2n-2. Take a; b 2 S such that [a; b] 6= 1. We assume |C S*
*(a)| < 2n-2,
because otherwise we are done. This gives |C S(a0ab)| > 2n - 2n-2 = 2n-1 + 2n-2*
*. So, we can
also assume a0ab 62 S.
If there exists a maximal elementary abelian subgroup E Gn such that a0ab 2*
* E and
|E \ S|=1, then by an argument similar to the one in Lemma 5.8, we have |S - CS*
*(a0ab)|
2n-1, which implies |C S(b)| 2n-2. So, assume contrary that for every maximal *
*elementary
abelian subgroup E G that includes the element a0ab, we have |S \E| 2. Now we*
* consider
the following cases:
Case 1: Suppose that there exists an element s 2 S such that a0abs 62 S. L*
*et E3 be
the subgroup generated by the elements a0ab and s, and the central element c 2 *
*Gn. Let E3
denote the set of maximal elementary abelian subgroups in Gn that include E3. B*
*y Lemma 2.2,
|E3| = t(Gn-2). Since E3\S = {s}, and |E\S| 2 for every E 2 E3, we have E\(S-{*
*s}) 6= ;.
Note that an element s02 S-{s} is included in t(Gn-3) maximal elementary abelia*
*n subgroups
in E3. So, there are at least t(Gn-2)=t(Gn-3) = 2n-2+1 elements in S -{s} which*
* are included
in a maximal elementary abelian subgroup E 2 E3. Since each of these elements w*
*ill commute
with s, we conclude that |C S(s)| 2n-2 + 2 2n-2.
Case 2: Suppose that for every s 2 C S(a0ab), abs 2 S, and there exists an e*
*lement x 2 S
such that [x; a0ab] 6= 1. Then, x commutes with either s or a0abs for every s *
*2 S. Since
C S(a0ab) can be written as disjoint union of X and a0abX for some X S, we obt*
*ain
|C S(x)| |C S(a0ab)|=2 > (2n-1 + 2n-2)=2 > 2n-2:
Case 3: Finally, we assume that for every s 2 C S(a0ab), a0abs 2 S, and S C*
* S(a0ab).
Repeting the argument in the proof of Lemma 5.8, we obtain that S = S0t a0abS0 *
*for some
S0 S such that S0is a representing set for a subgroup isomorphic to Gn-1. So, b*
*y induction
we obtain |C S(s)| 2n-2 for some s 2 S.
15
The proof of the proposition is complete. *
* |___|
7 Open Problems
We now would like to list some problems which we find interesting:
Problem 7.1. Show that the equality chl(G) = s(G) holds for all extra-special *
*2-groups.
Motivation for this problem is clear, since it will complete the calculation*
* of cohomology
lengths of extra-special 2-groups, and it will give the best possible upper bou*
*nds for cohomol-
ogy lengths of 2-groups obtained by using extra-special factor groups.
For odd primes, we do know that chl(G) p when G is an extra-special p-group*
* of type
(e) and (f), and we proved in this paper that if G is of type (d), then chl(G) *
*= s(G). So, for
odd primes what remains is the following calculation:
Problem 7.2. Calculate s(Gn) in terms of p and n, when Gn is an extra-special *
*p-group of
type (d).
As in the case of p = 2, for the calculation of s(Gn) for odd primes we need*
* a good
understanding of non-commuting structure of Gn. In particular, one would like t*
*o know how
big the invariant nc(Gn) is:
Problem 7.3. Calculate nc(Gn) in terms of p and n for extra-special p-groups o*
*f order p2n+1.
In a recent joint work with Jon Pakianathan [10], we study simplicial comple*
*xes associated
with the non-commuting structure of a group. Although this study has many diffe*
*rent aspects,
we hope that it will also provide a good understanding of the invariant nc(G), *
*and it will
eventually help us to solve some of these problems.
References
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*er Math.
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[2]D. Benson, Representations and cohomology II, Cambridge Studies in Advan*
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ematics 31, Cambridge University Press, Cambridge.
[3]N. Christofides, Graph theory, an algorithmic approach, Academic Press, *
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[4]L. Evens, The cohomology of groups, Oxford Mathematical Monographs, Clar*
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[5]J.W.P. Hirshfeld and J.A. Thas, General Galois Geometries, Oxford Scienc*
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[6]O. Kroll, A representation theoretical proof of a theorem of Serre, Arhu*
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* London
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[8]K. Metsch, The sets closest to ovoids in Q-(2n+1; q), Bull. Belg. Math. *
*Soc. 5 (1998),
389-392.
[9]T. Okuyama and H. Sasaki, Evens' norm maps and Serre's theorem on the co*
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[10]J. Pakianathan and E. Yalcin, Commuting and non-commuting complexes, pre*
*print,
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[11]L. Pyber, The number of pairwise non-commuting elements and the index of*
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in a finite group, J. London Math. Soc. (2) 35 (1987), 287-295.
[12]D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the*
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[13]J.P. Serre, Sur la dimension cohomologique des groupes profinis, Topolog*
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[14]J.P. Serre, Une relation dans la cohomologie des p-groupes, C.R. Acad. S*
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p-groups for an odd prime p, Math. Proc. Camb. Phil. Soc. 94 (1983), 449*
*-459.
Department of Mathematics
McMaster University
Hamilton, ON, Canada
L8S 4K1
E-mail address: yalcine@math.mcmaster.ca
17