Publicacions Ma em`a iques, Vol 39 (1995), 241–261.
CALCULATING THE GENUS
OF A DIRECT PRODUCT
OF CERTAIN NILPOTENT GROUPS
Pe e Hil on and Di k Sce enels
Abs ac
The Mislin genus G(N) o a fini ely gene a ed nilpo en g oup N
wi h fini e commu a o subg oup admi s an abelian g oup s uc-
u e. I Nsa isfies some addi ional condi ions —we say ha N
belongs o N1— we know exac ly he s uc u e o G(N). Con-
side ing a di ec p oduc N1×···×Nko g oups in N1 akes us
i ually always ou o N1. We he e calcula e he Mislin genus o
such a di ec p oduc .
1. In oduc ion.
By N0we deno e he class o fini ely gene a ed infini e nilpo en g oups
Nwi h fini e commu a o subg oup [N,N]. F om [1], [2] we know ha
he (Mislin) genus G(N), o N∈N
0, may hen be gi en he s uc u e
o a fini e abelian g oup. Mo eo e , i Nis a nilpo en g oup and we
conside he sho exac sequence 0 →TN →N→FN →0, whe e
TN is he o sion subg oup o Nand FN he o sion ee quo ien , hen
N∈N
0i and only i TN is fini e and FN is ee abelian o fini e ank.
I addi ionally
(1) TN is abelian;
(2) 0 →TN →N→FN →0 spli s on he igh , so ha Nis he
semidi ec p oduc o an ac ion ω:FN →Au (TN) o FN on
TN;
(3) he ac ion ωsa isfies ω(FN)⊆ZAu (TN), whe e Zdeno es he
cen e,
hen we say ha N∈N
1⊂N
0. No e ha (fini e) di ec p oduc s o
membe s o N1inhe i p ope ies (1) and (2) abo e, bu no , in gene al,
p ope y (3).
242 P. Hil on, D. Sce enels
Recall om [3] ha , gi en (1), (3) is equi alen o equi ing ha o
each ξ∈FN, he e exis s u∈Z, p ime o exp TN, such ha ξ·a=ua
o all a∈TN (TN is he e w i en addi i ely).
Now i is he heigh o ke ωin FN (meaning ha is he la ges
posi i e in ege msuch ha ke ω⊆mF N), hen we know om [3]
1.1. Theo em. G(N)∼
=(Z/ )∗/{±1}, o N∈N
1.
Mo eo e i is p o ed in [4] ha
1.2. Theo em. Fo N∈N
1wi h FN no cyclic, G(Nk)=0 o any
k≥1, whe e Nkis he k h di ec powe o N.
Now, i exp TN =n=pm1
1...p
ms
s,p1<p
2<··· <p
s,mi≥1, we
know ha mus ha e he o m =pλ1
1pλ2
2...p
λs
s, wi h 0 ≤λi<m
i
(i=1,2,... ,s); we w i e pλi
i ,i=1,2,... ,s. We also w i e T(N) o
he collec ion o p imes (p1,p
2,... ,p
s).
In [5] he au ho s calcula e G(Nk) o N∈N
1wi h FN cyclic and
k≥2, ob aining he ollowing heo em.
1.3. Theo em. Fo N∈N
1wi h FN cyclic and o any k≥2,we
ob ain G(Nk) om G(N)by ac o ing ou hose esidues mmod such
ha
m≡imod pλi
i,
i=±1,i=1,2,... ,s.
In his pape we will gene alize hese calcula ions o ob ain a esul o
he genus o a di ec p oduc , G(N1×···×Nk), whe e N1,... ,N
k∈N
1.
In he hi d sec ion we will show ha , i he di ec p oduc in ol es a
g oup Nj∈N
1wi h a non-cyclic o sion ee quo ien FNj, hen he
genus o he di ec p oduc is i ial. No e ha his is a gene aliza ion
o Theo em 1.2. In ac , we p o e
1.4. Theo em. Fo N1∈N
1wi h FN1no cyclic and N2∈N
0,we
ha e
G(N1×N2)=0.
In he case whe e he di ec p oduc only in ol es g oups
N1,N
2,... ,N
k∈N
1, all wi h a cyclic o sion ee quo ien FNi,an
impo an ole is played by he so-called gene a o s ha obs uc an iso-
mo phism. In he defini ion below, we w i e |a| o he o de o he
elemen ain some gi en g oup.
The genus o a di ec p oduc o nilpo en g oups 243
1.5. Defini ion. Le N1,N
2∈N
1and p∈T=T(N1×N2). Suppose
ha
(TN1)p=a1(1)⊕a2(1)⊕···⊕as1(1),
whe e exp(TN1)p=|a1(1)|≥|a2(1)|≥···≥|as1(1)|;
(TN2)p=a1(2)⊕a2(2)⊕···⊕as2(2),
whe e exp(TN2)p=|a1(2)|≥|a2(2)|≥···≥|as2(2)|.
Le 2 ≤x≤min(s1,s
2) + 1, and suppose ha
|a1(1)|=|a1(2)|,|a2(1)|=|a2(2)|,... ,|ax−1(1)|=|ax−1(2)|.
Then we say ha ax(1) obs uc s an isomo phism be ween (TN1)pand
(TN2)pi ei he |ax(1)|=|ax(2)|o x=s2+1 ≤s1. Simila ly we
speak o ax(2) obs uc ing an isomo phism. We call he o de o obs uc-
ion (o (TN1)p,(TN2)p) he maximum o he o de s o all gene a o s
o (TN1)p,(TN2)pobs uc ing an isomo phism. O cou se, he o de o
obs uc ion is independen o he choice o di ec sum decomposi ion o
(TN1)p,(TN2)p.
In he cou se o he ou h sec ion we will p o e ou main heo em,
namely,
1.6. Theo em. Le N1,N
2,... ,N
k∈N1wi h G(Ni)∼
=(Z/ i)∗/{±1}.
Se
= gcd( 1,... ,
k)=pλ1
1...p
λs
s.
Le FNi=ξiwi h ξi·a=uia o a∈TNi. Define P o be he se o
p ime di iso s po such ha he e a e dis inc , ∈{1,... ,k} o
which he ollowing condi ions hold:
(1) exp(TN )p= exp(TN )p;
(2) u ∈u ,u ∈u , whe e u ,u a e iewed as elemen s o
(Z/exp(TN )p)∗;
(3) On hose gene a o s o (TN )pand (TN )p ha obs uc an iso-
mo phism be ween hese wo o sion g oups, he ac ions o ξ ,ξ
a e i ial. This means ha
u ≡u ≡1modulo he o de o obs uc ion.
Then we ob ain G(N1×···×Nk) om (Z/ )∗by ac o ing ou he esidue
class o −1and hose esidues mmod such ha
m≡1modpλi
i o all pi/∈P
m≡1o −1modpλi
i o all pi∈P.
244 P. Hil on, D. Sce enels
No e ha his is indeed a gene aliza ion o Theo em 1.3. Fo i N1=
N2=···=Nk, hen Pwould consis o all p imes pidi iding , so ha
G(Nk) would be ob ained om (Z/ )∗by di iding ou hose esidues
mmod which a e cong uen o 1 o −1modpλi
i o all pi.
1.7. Co olla y. Assume u he ha =pλ. Then, wi h no u he
hypo hesis,
G(N1×···×Nk)∼
=(Z/ )∗/{±1}.
I is also in e es ing o no e ha he condi ion (2), namely, u ∈u ,
u ∈u is in ac equi alen o |u |=|u |, i he g oup (Z/exp(TN )p)∗
is cyclic. This g oup is indeed cyclic i p= 2. Howe e i p= 2 and
m≥3, he g oup (Z/2m)∗is no cyclic, and in his case we canno
eplace he gi en condi ion by he weake condi ion |u |=|u |, as Ex-
ample 4.4 will show.
We an icipa e ha he no ions o gene a o s obs uc ing an isomo -
phism and he o de o obs uc ion o an isomo phism may p o e o be
o in e es beyond he scope o his pape . No ice ha we only apply
hese no ions o g oups N1,N2such ha exp(TN1) = exp(TN2), since
we insis in Defini ion 1.5 ha x≥2.
2. Some p elimina y esul s.
Recall om [2], [3] he ollowing exac sequence (whe e N∈N
0)
T- Au Nθ
−→ (Z/e)∗/{±1}→G(N)→0.
He e T=T(N) is he se o p ime di iso s o n= exp TN,QN =
N/FZN,FZN being he ee cen e o N,e= exp QNab, and T-Au N
is he semig oup o sel T-equi alences o N. Recall also how θac s. Fo
any T-au omo phism ϕ,θ(ϕ) is he esidue class modulo ±1 o de ϕ,ϕ
being es ic ed o FZN. (In [1], [2] i is shown ha a T-au omo phism
sends FZN o i sel ). Mo eo e in [5] he au ho s show he ollowing.
2.1. Lemma. Le ϕ:N→Nbe an endomo phism. Then ϕinduces
ψ:FN →FN.I ϕ(FZN)⊆FZN, hen de (ϕ|FZN) = de ψ.
So, o a T-au omo phism ϕo N,θ(ϕ) is in ac he esidue class o
de ψ.Fo N∈N
0sa is ying condi ions (1) and (2) o N1, we also ha e
he ollowing ([5], [6]).
The genus o a di ec p oduc o nilpo en g oups 245
2.2. Lemma. An endomo phism ϕo Ninduces a commu a i e di-
ag am 0−−−−→TN −−−−→N−−−−→FN −−−−→0


α


ϕ

ψ
0−−−−→TN −−−−→N−−−−→FN −−−−→0
and ϕis a T-au omo phism i and only i αis an au omo phism and ψ
is a T-au omo phism.
2.3. Lemma.
(i) Fo all ξ∈FN and o all a∈TN, we ha e α(ξ·a)=ψ(ξ)·α(a).
(ii) Suppose ha a diag am
0−−−−→TN −−−−→N−−−−→FN −−−−→0


α

ψ
0−−−−→TN −−−−→N−−−−→FN −−−−→0
is gi en, such ha α(ξ·a)=ψ(ξ)·α(a), o all ξ∈FN and o all
a∈TN. Then we may find ϕ:N→Nmaking a commu a i e
diag am as in he p e ious lemma.
We call (i) abo e he compa ibili y condi ion.
3. The genus o a di ec p oduc , in ol ing a g oup in N1
wi h a non-cyclic o sion ee quo ien .
P oo o Theo em 1.4: Se T=T(N1)∪T(N2). Since N1×N2∈N
0,
we ha e he ollowing exac sequence:
T- Au (N1×N2)θ
−−−−→(Z/e)∗/{±1}−−−−→G(N1×N2)−−−−→0
whe e e= lcm(e1,e
2). We show ha we can ealize he esidue class o
any m, p ime o e, by some T-au omo phism φo N1×N2. In o he
wo ds we show ha o any mp ime o e, he e exis s a commu a i e
diag am
0−−−−→TN1×TN2−−−−→N1×N2−−−−→FN1×FN2−−−−→0
α



φ

ψ
0−−−−→TN1×TN2−−−−→N1×N2−−−−→FN1×FN2−−−−→0
whe e αis an au omo phism and φ,ψa e T-au omo phisms, such ha
de ψ=m.

246 P. Hil on, D. Sce enels
Choose a basis o FN1such ha
FN1=ξ1,ξ
2,... ,ξ
,ke ω1= 1ξ1,
2ξ2,... ,
ξ 
whe e = 1| 2|... | and ω1is he ac ion o FN1on TN1. Le
ξi·a=uia o a∈TN1. Rema k ha he o de o uimodulo exp(TN1)
is hen i. Now se
α=Id
TN
1×TN
2
and (in addi i e no a ion)





ψ(ξ1)=mξ1+lξ2,whe e l emains o be de e mined
ψ(ξj)=ξj(j=1)
ψ|FN2=Id
FN
2.
Then we ha e only o e i y he compa ibili y condi ion (Lemma 2.3) o
ξ1.Now
α(ξ1·a)=ψ(ξ1)·α(a) o all a∈TN1×TN2
i and only i
u1a=um
1ul
2a o all a∈TN1,
which is equi alen o
(3.1) um−1
1ul
2≡1 mod exp(TN1).
We now ha e one o he ollowing h ee possibili ies:
(1) I eis e en and 1is e en, hen mis odd, m−1 is e en and hus
um−1
1∈u2
1;
(2) I eis e en and 1is odd, hen u1∈u2
1, since 1is odd;
(3) I eis odd, hen 1is odd (because 1|e1|e) and u1∈u2
1, since
1is odd.
Mo eo e , by he same a gumen as in Theo em 1.1 o [4], we can show
ha in any case u2
1∈u2. Thus in each o he h ee cases i is clea ha
we can always sol e (3.1) o l. Mo eo e de ψ=m, which comple es
ou p oo .
4. The genus o a di ec p oduc o N1,... ,N
kin N1, each Ni
ha ing a cyclic o sion ee quo ien FNi.
Le N1,N
2,... ,N
k∈N
1wi h G(Ni)∼
=(Z/ i)∗/{±1}, ibeing defined
as in Sec ion 1. Se = gcd( 1,... ,
k)=pλ1
1...p
λs
sand se T=T(N1×
N2×···×Nk). Suppose ha , o p| and o i=1,... ,k,
(TNi)p=a1(i)⊕a2(i)⊕···⊕asi(i)
The genus o a di ec p oduc o nilpo en g oups 247
wi h exp(TNi)p=|a1(i)|≥|a2(i)|≥···≥|asi(i)|. Le FNi=ξiwi h
ξi·a=uia o a∈TNi.
To calcula e G(N1×···×Nk) o N1,... ,N
k∈N
1, we will use he
exac sequence
T- Au (N1×···×Nk)θ
−→ (Z/e)∗/{±1}→G(N1×···×Nk)→0,
which is alid, since clea ly N1×···×Nk∈N
0. In he ollowing wo
p oposi ions we will gi e a desc ip ion o im θ. F om hese we can hen
conclude how o use Theo em 1.6 o ob ain G(N1×···×Nk).
4.1. P oposi ion. Conside he ollowing commu a i e diag am:
0−→ TN1×···×TNk−→ N1×···×Nk−→ FN1×···×FNk−→ 0


α


ϕ

ψ
0−→ TN1×···×TNk−→ N1×···×Nk−→ FN1×···×FNk−→ 0
whe e αis an au omo phism and ϕ,ψa e T-au omo phisms. Le p| .
Le ψ(ξm)=k
j=1 βmjξj o m=1,... ,k, and le
αp(ai( ))=
s1

=1
α(1)
i( )a(1) +
s2

=1
α(2)
i( )a(2) +···+
sk

=1
α(k)
i( )a(k)
o ∈{1,... ,k}and i∈{1,... ,s
}.
Then he e exis s a bijec ion :{1,... ,k}−→{1,... ,k}:j−→ (j),
whe e (j)is he unique index such ha
pα1(j)
q( (j)) o some q∈{1,... ,s
(j)}.
Mo eo e , we also ha e
(1) exp(TNj)p= exp(TN (j))p;
(2) uj∈u (j),u (j)∈uj, wi h uj,u (j) iewed as elemen s o
(Z/exp(TNj)p)∗;
(No e ha , o cou se, (1) and (2) become i ial i j= (j).)
(3) u (j)≡u ( )≡1mod|α( )
q( (j))a( )| o all =j, o all &∈
{1,... ,s
}and o all q∈{1,... ,s
(j)}.
Fu he , i (j)=j o all j∈{1,... ,k}, hen de ψ≡1modpλ;
and i he e exis s j∈{1,... ,k}such ha (j)=j, hen
de ψ≡±1modpλ. In he la e case we need, mo eo e , he condi ion
uj≡u (j)≡1modulo he o de o obs uc ion o (TNj)p,(TN (j))p.
248 P. Hil on, D. Sce enels
P oo : The compa ibili y condi ion (Lemma 2.3) ells us ha
ψ(ξm)·α(ai( ))=α(ξm·ai( ))
o m∈{1,... ,k}, ∈{1,... ,k},i∈{1,... ,s
}. This yields he
ollowing.
I m= , hen


uβ 1
1α(1)
i( )a(1) +···+

uβ k
kα(k)
i( )a(k)
=

u α(1)
i( )a(1) +···+

u α(k)
i( )a(k).
I m= , hen


uβm1
1α(1)
i( )a(1) +···+

uβmk
kα(k)
i( )a(k)
=

α(1)
i( )a(1) +···+

α(k)
i( )a(k).
This means ha , o all m, , ∈{1,... ,k}, o all i∈{1,... ,s
}, and
o all &∈{1,... ,s
}:
I m= , hen uβ
α( )
i( )a( )=u α( )
i( )a( );
I m= , hen uβm
α( )
i( )a( )=α( )
i( )a( ).
Thus
I m= , hen uβ
≡u mod |α( )
i( )a( )|;(4.1)
I m= , hen uβm
≡1mod|α( )
i( )a( )|.(4.2)
We now asse ha
∀j∈{1,... ,k}∃! (j)∈{1,... ,k}such ha pα1(j)
q( (j)) o some q.
Indeed, since pdoes no di ide he de e minan o αp, he e ce ainly
exis s such a (j). And i we suppose ha he e exis , such ha
pα1(j)
i( ) o some i∈{1,... ,s
}
and
pα1(j)
i( ) o some i∈{1,... ,s
},
The genus o a di ec p oduc o nilpo en g oups 249
hen i ollows om (4.2) ha
∀m= u
βmj
j≡1mod|a1(j)|= exp(TNj)p
and
∀m= uβmj
j≡1mod|a1(j)|= exp(TNj)p.
I = , his would imply ha
∀mβ
mj ≡0mod( j)p,
whe e ( j)ps ands o he p-pa o j. Hence i would ollow ha de ψ=
de (βij)≡0mod( )p. Howe e , his is impossible, since pde ψ(ψ
being a T-au omo phism). The asse ion assu es us ha he ma ix o
αp, educed mod p, looks like
a1(j)α(TN (j))p























0
0
0
0
... ... .
.
.... ... ...
*
*
... ... .
.
.... ... ...
0
0
0
.
.
.
0























No e ha he abo e also implies ha exp(TNj)p≤exp(TN (j))p.
Thus we ha e se up a map :{1,... ,k}−→{1,... ,k}:j−→ (j).
We claim ha his map is a bijec ion. Indeed, i we suppose ha
(j)= (j)= , meaning ha pα1(j)
q( ) o some q∈{1,... ,s
}and
ha pα1(j)
q( ) o some q∈{1,... ,s
}, hen we would ge om (4.2)
ha
∀m= u
βmj
j≡1mod|a1(j)|
∀m= u
βmj
j≡1mod|a1(j)|,
and hus
∀m= β
mj ≡0mod( j)p
∀m= β
mj≡0mod( j)p.
256 P. Hil on, D. Sce enels
4.2. P oposi ion. Le m∈(Z/ )∗wi h m≡imod pλi
i, o all
i∈{1,... ,k}whe e i=1o −1. Addi ionally i i=−1, hen suppose
ha he e exis , ∈{1,... ,k}such ha = and
(1) exp(TN )p= exp(TN )p( ha is |a1( )|=|a1( )|);
(2) u ∈u ,u ∈u , whe e u ,u
a e iewed as elemen s o
(Z/exp(TN )p)∗;
(3) u ≡u ≡1modulo he o de o obs uc ion o (TN )p,(TN )p.
Then we can ealize m, ha is, [m]∈im θ.
P oo : We will cons uc an au omo phism α∈Au (TN1×···×TNk)
and a T-au omo phism ψ∈T- Au (FN1×···×FNk), which sa is y he
compa ibili y condi ion o Lemma 2.3, such ha de ψ≡mmod .I
will ollow ha any endomo phism ϕo (N1×···×Nk), compa ible wi h
αand ψ, will be a T-au omo phism ealizing m. We will de e mine α
comple ely, bu we will only de e mine he ma ix o ψmod .
Fix a pa icula pamong he p ime di iso s o and le
p1 1,p
2 2,... ,p
k k,p
λ .
Se 




ψ(ξ1)=β11ξ1+···+β1kξk
...
ψ(ξk)=βk1ξ1+···+βkkξk.
The idea is he ollowing. I m≡1modpλ, we will cons uc αp
as he iden i y on (TN1×···×TNk)pand he ma ix o
ψ, educed mod pλ, should look like he iden i y ma ix. I
m≡−1modpλ, hen αpshould map (TN )p o (TN )pand ice- e sa
as much as possible. This means ha we map he espec i e gene a o s
wi h he same o de ( o example a1( )and a1( )) on each o he . On he
gene a o s o (TN )pobs uc ing an isomo phism, and on la e gene -
a o s, we define αp o be he iden i y, and likewise o (TN )p. On all
o he p- o sion subg oups (TNj)p o j= , , we also define αp o be he
iden i y. The ma ix o ψ, educed mod pλ, will look like he iden i y
ma ix ou side he h and h columns. These wo columns con ain β
and β such ha u ≡uβ
,u ≡uβ
mod exp(TN )p. Then, as we
will show, de ψwill be cong uen o −1modpλ.
Case 1: m≡1modpλ.
Define αp=Id:(TN1×···×TNk)p→(TN1×···×TNk)pand le
βii ≡1modpi o all i∈{1,... ,k}
βij ≡0modpji j=i.

The genus o a di ec p oduc o nilpo en g oups 257
Case 2: m≡−1modpλ.
We hen know ha he e exis , ∈{1,... ,k}, = such ha
(1) exp(TN )p= exp(TN )p( ha is |a1( )|=|a1( )|);
(2) u ∈u ,u ∈u  iewed as elemen s o (Z/exp(TN )p)∗;
(3) u ≡u ≡1 modulo he o de o obs uc ion o (TN )p,(TN )p.
Define αp:(TN1×···×TNk)p→(TN1×···×TNk)pas ollows
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αp= Id ou side (TN ×TN )p;
αp= Id o he gene a o s o (TN )pand (TN )p
obs uc ing an isomo phism and o la e gene a o s;
αp(aj( ))=aj( )and αp(aj( ))=aj( )
o he o he gene a o s o (TN ×TN )p;
and le
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βii ≡1modpi o i= ,
β ≡0modp
β ≡0modp
β and β be chosen such ha
u ≡uβ
,u
≡uβ
mod exp(TN )p
(which is always possible, by hypo hesis)
βij ≡0modpjo he wise .
Rema k ha in bo h cases we can sol e all he cong uences
(by he Chinese Remainde Theo em) and ha βij will be
de e mined mod j, so ha he en ies o he ma ix o ψwill be de-
e mined mod gcd( 1,... ,
k)= . We will now check ha αand ψ,as
cons uc ed abo e, sa is y he compa ibili y condi ion (Lemma 2.3).
Case 1: m≡1modpλ
α(ξs·aq)=ψ(ξs)·α(aq)(aq∈TNq)(q=s)
⇐⇒ aq=uβsq
qaq,and he la e holds since βsq ≡0modpq
α(ξs·as)=ψ(ξs)·α(as)
⇐⇒ usas=uβss
sas,and he la e holds since βss ≡1modps.
Case 2: m≡−1modpλ
I {q,s}={ , }, we ge simila equa ions o hose abo e. I {q,s}=
{ , }, we ha e o gene a o s aj( ),aj( ) ha a e mapped unde αon
258 P. Hil on, D. Sce enels
each o he :
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ aj( )=aj( )
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ u aj( )=uβ
aj( )
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ u aj( )=uβ
aj( )
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ aj( )=aj( ),
and he la e ela ions all hold.
I {q,s}={ , }, we ha e o gene a o s aj( ),aj( )on which αis
defined as he iden i y ( ha is, gene a o s obs uc ing an isomo phism
o la e gene a o s):
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ aj( )=uβ
aj( )
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ u aj( )=aj( )
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ u aj( )=aj( )
α(ξ ·aj( ))=ψ(ξ )·α(aj( ))⇐⇒ aj( )=uβ
aj( ),
and he la e ela ions all hold, by (3) abo e.
Finally we look a de ψ. Fo each p| ,weha e
de ψ= de (βij)≡1modpλi m≡1modpλ
−β β mod pλi m≡−1modpλ.
Howe e in he second case we know
u ≡uβ
≡(uβ
)β mod exp(TN )p.
F om his i ollows ha β β ≡1modp =p , so in ei he case we
ha e
de ψ≡mmod pλ.
Thus
de ψ≡mmod ,
which concludes he p oo o P oposi ion 4.2. Wi h hese wo p oposi-
ions ou main esul , Theo em 1.6, is es ablished.
We now gi e an example o how one can use Theo em 1.6 o calcula e
he genus o a di ec p oduc o g oups in N1.
4.3. Example.
Le N1∈N
1wi h TN1=Z/9⊕Z/49, FN1=ξ1and ξ1·a=22a
o all a∈TN1.Sou1=1+3·7 and 1=3·7 = 21. Le N2∈N
1
The genus o a di ec p oduc o nilpo en g oups 259
wi h TN2=Z/9⊕Z/3⊕Z/343 and FN2=ξ2whe e ξ2·b= 148b o
all b∈TN2.Sou2=1+3·72and 2=3·7 = 21. Thus = 21. Then
P={3}(see Theo em 1.6). No e ha 22 ≡148 ≡1 mod 3 (3 being
he o de o obs uc ion o (TN1)3,(TN2)3), bu plainly 7 ∈ P.Thuswe
ha e o ac o ou o (Z/21)∗ he esidue classes o 1,−1 and o hose m
such ha m≡1mod7,m≡±1 mod 3. This means ac o ing ou he
g oup Hgene a ed by {−1,m}, whe e m≡1mod7,m≡−1mod3.
Thus H=−1, 8.Thus
G(N1×N2)∼
=(Z/21)∗/H ∼
=Z/3.
O cou se, we can explici ly desc ibe he g oups in he genus o N1×N2,
using he desc ip ions o he g oups in G(N), o N∈N
1, gi en in [3].
Finally we gi e he p omised example o show ha he condi ion u ∈
u ,u ∈u canno be eplaced by he weake condi ion |u |=|u |
in Theo em 1.6. Tha is, we will gi e an example whe e 2 ∈ P, al hough
he p ime 2 sa isfies (1), (3) and he weake o m o (2); and whe e
he compa ibili y condi ion o Lemma 2.3 excludes he condi ion de ψ≡
−1mod2
λ.
4.4. Example.
Le
N1=x, y |x16 =1,yxy−1=x3
N2=x, y |x16 =1,yxy−1=x5.
Then
N1∈N
1,wi h TN1=Z/16 = a1,FN
1=Z=ξ1and ξ1·a1=3a1
N2∈N
1,wi h TN2=Z/16 = a2,FN
2=Z=ξ2and ξ2·a2=5a2.
Mo eo e 1=4 and 2=4, so =gcd( 1,
2) = 4. Le ψ∈T- Au (FN1×
FN2) be gi en by
ψ(ξi)=βi1ξ1+βi2ξ2, o i=1,2,
and le α∈Au (TN1×TN2) be gi en by
α(aj)=αj1a1+αj2a2, o j=1,2.
260 P. Hil on, D. Sce enels
Exp essing he condi ion α(ξi·aj)=ψ(ξi)·α(aj) o i,j=1,2 yields
he ollowing equa ions :
3α11a1+3α12a2=α113β11 a1+α125β12 a2
(1)
α21a1+α22a2=α213β11 a1+α225β12 a2
(2)
α11a1+α12a2=α113β21 a1+α125β22 a2
(3)
5α21a1+5α22a2=α213β21 a1+α225β22 a2
(4)
Now, we ha e a leas one o wo cases; ei he 2 α11 o 2 α21.I
2α11, hen we need 2|α21 (o he wise (1) and (2) con adic ) so ha
2α22 (because 2 de α) and so 2|α12 (o he wise (3) and (4) con adic ).
Howe e , in his case we ob ain
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β11 ≡1mod4
β12 ≡0mod4
β21 ≡0mod4
β22 ≡1mod4
So ha de ψ≡1mod4.
I 2 α21, hen analogously 2|α11,2α12 and 2|α22. He e we need
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β11 ≡0mod4
β22 ≡0mod4
3≡5β12 mod 16
5≡3β21 mod 16
The wo las cong uences howe e ha e no solu ion. We hus can con-
clude ha o each ψ∈T- Au (FN1×FN2) we ha e de ψ≡1mod4;
and de ψ≡−1 mod 4 is impossible.
O cou se, Co olla y 1.7 gi es us he simple o mula o G(N1×N2) in
his case, since only he p ime 2 is in ol ed; and he alue o G(N1×N2)
is unaffec ed by whe he we can find ψ∈T-Au (N1×N2) wi h de ψ≡
−1 mod 4. To ob ain a coun e example o he s a emen o Theo em 1.6
wi h he weake e sion o condi ion (2), we need o complica e ou
Example 4.4 by in ol ing ano he p ime pas a ac o o , in addi ion
o he p ime 2, and a anging ha p/∈P. We would he eby ob ain an
example in which all he hypo heses o Theo em 1.6 we e e ified, excep
ha condi ion (2) is eplaced by he weake e sion, bu he conclusion
o he heo em is alse.
The genus o a di ec p oduc o nilpo en g oups 261
Re e ences
1. G. Mislin,“Nilpo en g oups wi h fini e commu a o subg oups,”
Lec u e No es in Ma h. 418, Sp inge -Ve lag, 1974, pp. 103–120.
2. P. Hil on and G. Mislin, On he genus o a nilpo en g oup wi h
fini e commu a o subg oup, Ma h. Z. 146 (1976), 201–211.
3. C. Casacube a and P. Hil on, Calcula ing he Mislin genus o
a ce ain amily o nilpo en g oups, Comm. in Alg. 19(7) (1991),
2051–2069.
4. P. Hil on and C. Schuck, On he s uc u e o nilpo en g oups o
a ce ain ype, Topological Me hods in Nonlinea Analysis, Jou nal
o he Juliusz Schaude Cen e 1(1993), 323–327.
5. P. Hil on and C. Schuck, Calcula ing he genus o ce ain nilpo-
en g oups, Bull. Mex. Ma h. Soc. 37 (1992), 263–269.
6. P. Hil on, Non-cancella ion p ope ies o ce ain fini ely p esen -
ed g oups, Quaes iones Ma h. 9(1986), 281–292.
Pe e Hil on:
Depa men o Ma hema ical Sciences
S a e Uni e si y o New Yo k
Bingham on
New Yo k 13902-6000
U.S.A.
Di k Sce enels:
K.U. Leu en
Fakul ei We enschappen
Depa emen Wiskunde
Celes ijnenlaan 200B
B-3001 He e lee
BELGIUM
Rebu el 5 de Se emb e de 1994