Fast decomposition of p-groups in the Roquette category, for p > 2

Serge Bouc

Abstract : Let p be a prime number. In [9], I introduced the Roquette category R_{p} of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_{p}, every finite p-group P admits a canonical direct summand ∂P, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups.

In this note, I would like to describe a fast algorithm to obtain such a decomposition, when p is odd.

AMS Subject classification : 18B99, 19A22, 20C99, 20J15. Keywords : p-group, Roquette, rational, biset, genetic.

1. Introduction

Let p be a prime number. The Roquette category R_{p} of finite p-groups, introduced in [9], is an additive tensor category with the following properties :

• Every finite p-group can be viewed as an object of R_{p}. The tensor product of two finite p-groups P and Q in R_{p} is the direct product P × Q.

• In R_{p}, any finite p-group has a direct summand ∂P, called the edge

of P, such that

P ≅ ⊕_{N ⊴ P} ∂(P/N) .

Moreover, if the center of P is not cyclic, then ∂P = 0.

• In R_{p}, every finite p-group P decomposes as a direct sum

P ≅ ⊕_{R∈S} ∂R ,

where S is a finite sequence of Roquette groups, i.e. of p-groups of normal p-rank 1, and such a decomposition is essentially unique. Given the group P, such a decomposition can be obtained explicitly from the knowledge of a genetic basis of P.

• The tensor product ∂P × ∂Q of the edges of two Roquette p-groups P and Q is isomorphic to a direct sum of a certain number ν_{P,Q} of copies of the edge ∂(P ⋄ Q) of another Roquette group (where both ν_{P,Q} and P ⋄ Q are known explicitly.

• The additive functors from R_{p} to the category of abelian groups are exactly the rational p-biset functors introduced in [4].

The latter is the main motivation for considering this category : any structural result on R_{p} will provide for free some information on such rational functors for p-groups, e.g. the representation functors R_{K}, where K is a field of characteristic 0 (see [2], [3], and L. Barker’s article [1]), the functor of units of Burnside rings ([6]), or the torsion part of the Dade group ([5]).

The decomposition of a finite p-group P as a direct sum of edges of Roquette p-groups can be read from the knowledge of a genetic basis of P. The problem is that the computation of such a basis is rather slow, in general. For most purposes however, the full details encoded in a genetic basis are useless, and it would be enough to know the direct sum decomposition.

Hence it would be nice to have a fast algorithm taking any finite p-group P as input, and giving its decomposition as direct sum of edges of Roquette groups in the category R_{p}. This note is devoted to the description of such an algorithm, when p > 2.

2. Rational p-biset functors

2.1. Recall that the characteristic property of the edge ∂P of a finite p-group in the Roquette category R_{p} is that for any rational p-biset functor F

∂F (P) = F̂(∂P) ,

where ∂F (P) is the faithful part of F (P), and F̂ denotes the extension of F to R_{p}. Also recall the following criterion ([7], Theorem 3.1):

2.2. Theorem : Let p be a prime number, and F be a p-biset functor. Then the following conditions are equivalent:

1. The functor F is a rational p-biset functor.

2. For any finite p-group P, the following conditions hold:

• if the center of P is non cyclic, then ∂F (P) = {0}.

• if E ⊴ P is a normal elementary abelian subgroup of rank 2, and if Z ≤ E is a central subgroup of order p of P, then the map

Res_{C_{P} (E)}^{P} ⊕ Def_{P/Z}^{P} : F (P) → F (C_{P} (E)) ⊕ F (P/Z)

is injective.

2.3. Let K be a commutative ring in which p is invertible. When P is a finite group, denote by CF_{K}(P) the K-module of central functions from P to K. The correspondence sending a finite p-group P to CF_{K}(P) is a rational p-biset functor:

2.4. Proposition : If P and Q are finite p-groups, if U is a finite (Q, P)- biset, and if f ∈ CF_{K}(P), define a map CF_{K}(U) : CF_{K}(P) → CF_{K}(Q) by

∀s ∈ Q, CF_{K}(U)(f)(s) = \frac{1}{|P |} ∑_{{u∈U, x∈P \atop su=ux}} f(x) .

With this definition, the correspondence P ↦ CF_{K}(P) becomes a rational p-biset functor, denoted by CF_{K}.

Proof : A straightforward argument shows that CF_{K}(U)(f) is indeed a central function on Q, hence the map CF_{K}(U) is well defined. It is also clear that this map only depends on the isomorphism class of the biset U, and that for any two finite (H, G)-bisets U and U′, we have

CF_{K}(U ⊔ U′) = CF_{K}(U) + CF_{K}(U′) .

Moreover if U is the identity biset at P, i.e. if U = P with biset structure given by left and right multiplication, then for f ∈ CF_{K}(P) and s ∈ P

CF_{K}(U)(f)(s) = \frac{1}{|P |} ∑_{{u∈U, x∈P \atop su=ux}} f(x) = \frac{1}{|P |} ∑_{u∈P} f(s^{u}) = f(s) ,

hence CF_{K}(U) is the identity map.

Now if R is a third finite p-group, and V is a finite (R, Q)-biset, then for any t ∈ R, setting λ = CF_{K}(V) ∘ CF_{K}(U)(f)(t), we have that

λ = \frac{1}{|Q|} ∑_{{v∈V, s∈Q \atop tv=vs}} \frac{1}{|P |} ∑_{{u∈U, x∈P \atop su=ux}} f(x)

= \frac{1}{|Q||P |} ∑_{(v,u)∈V ×U ; {s∈Q, x∈P \atop tv=vs, su=ux}} f(x)

= \frac{1}{|Q||P |} ∑_{{(v,u)∈V ×U, x∈P \atop t(v,_{Q}u)=(v,_{Q}u)x}} |\{s ∈ Q | tv = vs, su = ux\}| f(x)

λ = \frac{1}{|Q||P |} ∑_{{(v,_{Q}u)∈V ×_{Q}U, x∈P \atop t(v,_{Q}u)=(v,_{Q}u)x}} |Q : Q_{v} ∩_{u}P ||Q_{v} ∩_{u}P | f(x)

= \frac{1}{|P |} ∑_{{(v,_{Q}u)∈V ×_{Q}U, x∈P \atop t(v,_{Q}u)=(v,_{Q}u)x}} f(x) = CF_{K}(V ×_{Q} U)(f)(t) .

Hence CF_{K}(V) ∘ CF_{K}(U) = CF_{K}(V ×_{Q} U), and CF_{K} is a p-biset functor.

To prove that this functor is rational, we use the criterion given by Theorem 2.2. Suppose first that the center Z(P) of P is non-cyclic. Let E denote the subgroup of Z(P) consisting of elements of order at most p. Then saying that ∂CF_{K}(P) = {0} amounts to saying that for any f ∈ CF_{K}(P), the sum

S = ∑_{Z≤E} μ(1, Z)Inf_{P/Z}^{P}Def_{P/Z}^{P}f

is equal to 0, where μ denotes the Möbius function of the poset of subgroups of P (or of E). Equivalently, for any s ∈ P

S(s) = ∑_{Z≤E} μ(1, Z)\frac{1}{|P |} ∑_{{aZ∈P/Z, x∈P \atop saZ=aZx}} f(x) = 0 .

This also can be written as

S(s) = ∑_{Z≤E} μ(1, Z)\frac{1}{|P ||Z|} ∑_{{a∈P, x∈P \atop saZ=aZx}} f(x)

= \frac{1}{|P |} ∑_{Z≤E} \frac{μ(1,Z)}{|Z|} ∑_{a∈P, z∈Z} f(s^{a}.z)
= \frac{1}{|P |} ∑_{Z≤E} \frac{μ(1,Z)}{|Z|} ∑_{a∈P, z∈Z} f((sz)^{a})
= ∑_{Z≤E} \frac{μ(1,Z)}{|Z|} ∑_{z∈Z} f(sz)
= ∑_{z∈E} ( ∑_{z∈Z≤E} \frac{μ(1,Z)}{|Z|})f(sz) .

2.5. Lemma : Let E be an elementary abelian p-group of rank at least 2. Then for any z ∈ E

∑_{z∈Z≤E} \frac{μ(1,Z)}{|Z|} = 0 .

Proof : For z ∈ E, set σ(z) = ∑_{z∈Z≤E} \frac{μ(1,Z)}{|Z|}. Assume first that z ≠ 1, i.e.

|z| = p. If Z ∋ z is elementary abelian of rank r, then μ(1, Z) = (−1)^{r}p^{\binom{r}{2}}, hence \frac{μ(1,Z)}{|Z|} = (−1)^{r}p^{\binom{r−1}{2}−1} = − \frac{1}{p} μ(1, Z/<z>). Hence setting \overline{Z} = Z/<z>

and \overline{E} = E/<z>,

σ(z) = − \frac{1}{p} ∑_{1≤\overline{Z}≤\overline{E}} μ(1, \overline{Z}) = 0 ,

since |\overline{E}| > 1. Now

∑_{z∈E} σ(z) = σ(1) + ∑_{e∈E−\{1\}} σ(z) = ∑_{z∈Z} ∑_{z∈Z≤E} \frac{μ(1,Z)}{|Z|} = ∑_{1≤Z≤E} μ(1, Z) = 0

hence σ(1) = 0, completing the proof of the lemma.

It follows that S(s) = 0, hence S = 0, as was to be shown.

For the second condition of Theorem 2.2, suppose that E is a normal elementary abelian subgroup of P of rank 2, and that Z is a central subgroup of P of order p contained in E. Let f ∈ CF_{K}(P) which restricts to 0 to C_{P} (E), and such that

∀sZ ∈ P/Z, (Def_{P/Z}^{P}f)(sZ) = \frac{1}{|P |} ∑_{z∈Z} f(sz) = 0 .

Thus f(s) = 0 if s ∈ C_{P} (E). Assume that s ∉ C_{P} (E). Then for e ∈ E, the commutator [s, e] lies in Z. Moreover the map e ∈ E ↦ [s, e] ∈ Z is surjective. it follows that for any z ∈ Z, there exists e ∈ E such that s^{e} = sz. Thus f(sz) = f(s^{e}) = f(s). Hence Def_{P/Z}^{P}f(s) = f(s) = 0. Hence f = 0, as was to be shown.

3. Action of p-adic units

Let ℤ_{p} denote the ring of p-adic integers, i.e. the inverse limit of the rings ℤ/p^{n}ℤ, for n ∈ ℕ − {0}. The group of units ℤ_{p}^{×} is the inverse limits of the unit groups (ℤ/p^{n}ℤ)^{×}, and it acts on the functor CF_{K} in the following way: if ζ ∈ ℤ_{p}^{×} and P is a finite p-group, choose an integer r such that p^{r} is a multiple of the exponent of P, and let ζ_{p^{r}} denote the component of ζ in (ℤ/p^{r}ℤ)^{×}. For f ∈ CF_{K}(P), define ζ̂_{P} (f) ∈ CF_{K}(P) by

∀s ∈ P, ζ̂_{P} (f)(s) = f(s^{ζ_{p^{r}}}) .

Then clearly ζ̂_{P} (f) only depends on ζ, and this gives a well defined map

ζ̂_{P} : CF_{K}(P) → CF_{K}(P) .

One can check easily (see [8] Proposition 7.2.4 for details) that if Q is a finite p-group, and U is a finite (Q, P)-biset, then the square

\xymatrix{CF_{K}(P) \ar[r]^{ζ̂_{P}} \ar[d]_{CF_{K}(U)} & CF_{K}(P) \ar[d]^{CF_{K}(U)} \\ CF_{K}(Q) \ar[r]^{ζ̂_{Q}} & CF_{K}(Q)}

is commutative. In other words, we have an endomorphism ζ̂ of the functor CF_{K}. It is straightforward to check that for ζ, ζ′ ∈ ℤ_{p}^{×}, we have ζ̂ζ′ = ζ̂∘ ζ̂′, and that ˆ1 is the identity endomorphism of CF_{K}. So this yields an action of the group ℤ_{p}^{×} on CF_{K}.

It follows in particular that when n ∈ ℕ − {0}, and P is a finite p-group, if we set

F_{n}(P) = {f ∈ CF_{K}(P) | ∀s ∈ P, f(s^{1+p^{n}}) = f(s)} ,

then the correspondence P ↦ F_{n}(P) is a subfunctor of CF_{K}: indeed F_{n} is the subfunctor of invariants by the element 1 + p^{n} of ℤ_{p}^{×}.

It follows that F_{n} is a rational p-biset functor, for any n ∈ ℕ − {0}, hence it factors through the Roquette category R_{p}. In particular, for any finite p-group P, if P splits as a direct sum

P ≅ ⊕_{R∈S} ∂R

of edges of Roquette groups in R_{p}, then there is an isomorphism

F_{n}(P) ≅ ⊕_{R∈S} ∂F_{n}(R) .

3.1. Notation : For a finite p-group P, and an integer n ∈ ℕ − {0}, let l_{n}(P) denote the number of conjugacy classes of elements s of P such that s^{1+p^{n}} is conjugate to s in P. Also set l_{0}(P) = 1.

With this notation, for any finite p-group P, and any n ∈ ℕ − {0}, the K-module F_{n}(P) is a free K-module of rank l_{n}(P). In particular, if P = C_{p^{m}} is cyclic of order p^{m}, then F_{n}(P) has rank l_{n}(P) = p^{min(m,n)}. Thus if m > 0, then ∂F_{n}(C_{p^{m}}) has rank p^{min(m,n)} − p^{min(m−1,n)}, since C_{p^{m}} ≅ ∂C_{p^{m}} ⊕ C_{p^{m−1}} in R_{p}.

3.2. Theorem : Assume that a p-group P splits as a direct sum

P ≅ 1 ⊕ ⊕_{m=1}^{∞} a_{m}∂C_{p^{m}}

of edges of cyclic groups in the Roquette category R_{p}, where a_{m} ∈ ℕ. Then

∀m ≥ 1, a_{m} = \frac{l_{m}(P) − l_{m−1}(P)}{p^{m−1}(p − 1)} .

Proof : For any n ∈ ℕ − {0}, we have

l_{n}(P) = 1 + ∑_{m=1}^{∞} a_{m}(p^{min(m,n)} − p^{min(m−1,n)}) = 1 + ∑_{m=1}^{n} a_{m}(p^{m} − p^{m−1}) .

For n ∈ ℕ − {0}, this gives l_{n}(P) − l_{n−1}(P) = a_{n}(p^{n} − p^{n−1}).

3.3. Corollary : Suppose p > 2. If P is a finite p-group, then

P ≅ 1 ⊕ ⊕_{m=1}^{∞} \frac{l_{m}(P)−l_{m−1}(P)}{p^{m−1}(p−1)} ∂C_{p^{m}}

in the Roquette category R_{p}.

Proof : Indeed for p odd, all the Roquette p-groups are cyclic, hence the assumption of Theorem 3.2 holds for any P.

Appendix

3.1. A GAP function : The following function for the GAP software ([10]) computes the decomposition of p-groups for p > 2, using Corollary 3.3:

#
# Roquette decomposition of an odd order p-group g
# output is a list of pairs of the form [p^n,a_n]
# where a_n is the number of summands of g
# isomorphic to the edge of the cyclic group of order p^n
#

roquette_decomposition:=function(g)
local prem,cg,s,i,x,y,z,pn,u;
    if IsTrivial(g) then return [[1,1]];fi;
    prem:=PrimeDivisors(Size(g));
    if Length(prem)>1 then
        Print("Error : the group must be a p-group\n");
        return fail;
    fi;
    prem:=prem[1];
    if prem=2 then
        Print("Error : the order must be odd\n");
        return fail;
    fi;
    cg:=ConjugacyClasses(g);
    s:=[];
    for i in [2..Length(cg)] do
        x:=cg[i];
        y:=Representative(x);
        pn:=1;
        u:=y;
        repeat
            pn:=pn*prem;
            u:=u^prem;
            z:=y*u;
        until z in x;
        Add(s,pn);
    od;
    s:=Collected(s);
    s:=List(s,x->[x[1],x[2]*prem/(prem-1)/x[1]]);
    s:=Concatenation([[1,1]],s);
    return s;
end;

3.2. Example :

gap> l:=AllGroups(81);;
gap> for g in l do
> Print(roquette_decomposition(g),"\n");
> od;
[ [ 1, 1 ], [ 3, 1 ], [ 9, 1 ], [ 27, 1 ], [ 81, 1 ] ]
[ [ 1, 1 ], [ 3, 4 ], [ 9, 12 ] ]
[ [ 1, 1 ], [ 3, 7 ], [ 9, 3 ] ]
[ [ 1, 1 ], [ 3, 7 ], [ 9, 3 ] ]
[ [ 1, 1 ], [ 3, 4 ], [ 9, 3 ], [ 27, 3 ] ]
[ [ 1, 1 ], [ 3, 4 ], [ 9, 4 ] ]
[ [ 1, 1 ], [ 3, 8 ] ]
[ [ 1, 1 ], [ 3, 5 ], [ 9, 1 ] ]
[ [ 1, 1 ], [ 3, 5 ], [ 9, 1 ] ]
[ [ 1, 1 ], [ 3, 5 ], [ 9, 1 ] ]

[ [ 1, 1 ], [ 3, 13 ], [ 9, 9 ] ]
[ [ 1, 1 ], [ 3, 16 ] ]
[ [ 1, 1 ], [ 3, 16 ] ]
[ [ 1, 1 ], [ 3, 13 ], [ 9, 1 ] ]
[ [ 1, 1 ], [ 3, 40 ] ]

For example, the group on line 6 of the previous list, isomorphic to the semidirect product C_{27} ⋊ C_{3}, is isomorphic to 1 ⊕ 4∂C_{3} ⊕ 4∂C_{9} in R_{3}.

References

[1] L. Barker. Rhetorical biset functors, rational p-biset functors, and their semisimplicity in characteristic zero. J. of Algebra, 319(9):3810–3853, 2008.

[2] S. Bouc. Foncteurs d’ensembles munis d’une double action. J. of Algebra, 183(0238):664–736, 1996.

[3] S. Bouc. The functor of rational representations for p-groups. Advances in Mathematics, 186:267–306, 2004.

[4] S. Bouc. Biset functors and genetic sections for p-groups. J. of Algebra, 284(1):179–202, 2005.

[5] S. Bouc. The Dade group of a p-group. Inv. Math., 164:189–231, 2006.

[6] S. Bouc. The functor of units of Burnside rings for p-groups. Comm. Math. Helv., 82:583–615, 2007.

[7] S. Bouc. Rational p-biset functors. J. of Algebra, 319:1776–1800, 2008.

[8] S. Bouc. Biset functors for finite groups, volume 1990 of Lecture Notes in Mathematics. Springer, 2010.

[9] S. Bouc. The Roquette category of finite p-groups. preprint,

   http://fr.arxiv.org/abs/1111.3469, 2011.

[10] The GAP Group. GAP – Groups, Algorithms, and Programming, Ver-

   sion 4.6.3, 2013. (http://www.gap-system.org).

Serge Bouc - CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens Cedex 01 - France.

email :  serge.bouc@u-picardie.fr
web   :  http://www.lamfa.u-picardie.fr/bouc/