NAGASAKI, IKUMITSU

Nagasaki, Ikumitsu

Institutional Repositories DataBase (IRDB)

Homotopy representation groups and Swan subgroups

Osaka City University Repository

Nagasaki, I.
Osaka J. Math.
24 (1987), 253-261
HOMOTOPY REPRESENTATION GROUPS AND
SWAN SUBGROUPS
IKUMITSU NAGASAKI
(Received January 21, 1986)
0. Introduction
Let G be a finite group. A finite dimensional G-C£F-complex X is called
a homotopy representation of G if the i^-fixed point set XH is homotopy equiva-
lent to a (dim XH)-dimensional sphere or the empty set for each subgroup H
of G. Moreover if X is G-homotopy equivalent to a finite G-C W-complex,
then X is called a finite homotopy representation of G and if X is G-homotopy
equivalent to a unit sphere of a real representation of G, then X is called a linear
homotopy representation of G. T. torn Dieck and T. Petrie defined homo-
topy representation groups in order to study homotopy representations. Let
F+(G, h°°) be the set of G-homotopy types of homotopy representations. We
define the addition on V+(G, h°°) by the join and so V+(G> h°°) becomes a semi-
group. The Grothendieck group of V+(G, h°°) is denoted by V(G, h°°) and
called the homotopy representation group. A similar group F(G, h) [resp.
V(G, /)] can be defined for finite [resp. linear] homotopy representations.
Let (j>(G) denote the set of conjugacy classes of subgroups of G and C(G)
the ring of functions from <f>(G) to integers. For a homotopy representation
Xy the dimension function DimX in C{G) is defined by (Dim JY)(i/)=dim Xs
+ 1. (If XH is empty, then we set dimX*= — 1.) Then
Dim X* Y = Dim X+Dim Y
for any two homotopy representations. ("*" means the join.) Hence one
can define the homomorphism
Dim: F(G, X) -* C(G) (X = A~, h or /)
by the natural way. The kernel of Dim is denoted by v(G, X). torn Dieck and
Petrie proved that v(G, X) is the torsion group of V(G> X) and
(0.1) v(G, /*")^Picn(G),
where Pic Q,(G) is the Picard group of the Burnside ring H(G).
There are the natural homomorphisms
254 I. NAGASAKI
(Q2) jG:v(G,l)->v(G,h)
{
 '
 ]
 kG: v(G, h) -> v(G, h").
The homomorphisms jG and kG are injective in general and hence we often
regard v(G, I) and v(G, h) as the subgroups of v(G, h°°) via these injective homo-
morphisms. We prove the following result in Section 2.
Theorem A. The homomorphism kG is an isomorphism if and only if the
Swan subgroup T(G) vanishes.
The definition and properties of the Swan subgroup are mentioned in
Section 1. The finite groups with T(G)=0 are studied by Miyata and Endo
[7]. The Swan subgroups play an important role in the computation of v
(G, h). In fact, the computation of v(G, h) for an abelian group G is com-
pletely reduced to the computation of the Swan subgroups. By computing
the Swan subgroups of some groups, we prove the following result in Section 3.
Theorem B. Suppose that G is an abelian group. Then jG is an isomor-
phism if and only if G is isomorphic to one of the following groups.
( i ) C a cyclic group
(ii) G2 an abelian 2-group
(iii) G3 an abelian Z-group
(iv)
(v)
(vi) (Z/2)nx(ZI3)m
In Section 4 we determine the finite groups with v(G, A°°)=0 by using the
results in Section 2.
Theorem C. The group v(G, h°°) vanishes if and only if G is isomorphic to
one of the groups'.
Z\n (ii=l, 2, 3, 4 or 6), D(2n) (n=2y 3, 4 or 6), A4, S4.
Here D(2n) denotes the dihedral group of order 2n and Sn\A^\ denotes the
symmetric [alternating] group on n symbols.
1. The Swan subgroup
In this Section, we shall summarize the well-known results on the Swan
subgroup. Let 2 G be the sum of elements of G in the integral group ring
ZG and [r, S G ] be the left ideal generated by r and SG> where r^Z is prime
to the order \G\ of G. The ideal [r, ]T]G] is projective as a Z'G-module. Hence
[f> S G ] decides the element of the reduced projective class group KQ(ZG). From
[9], a homomorphism
HOMOTOPY REPRESENTATION GROUPS 255
is defined by SG(r)=([ry SG]), where Z\ \ G | * is the unit group of Z\ \ G |. We
put <G)=(Z/ |G |*) /±1 . Since SG(±1)=O, SG induces
which is called the Swan homomorphism. The image of SG is called the Swan
subgroup of G and denoted by T(G).
The following results are well-known.
Theorem 1.1 ([9]). If G is a cyclic group, then T(G)=0.
Theorem 1.2 ([11]).
( i ) T(G) is a quotient group of u(G).
(ii) Iff: G->G' is a surjective homomorphism , then the natural map fi[0(ZG)
->R0(ZG') sends ([r, 2G]) to ([r, IV]) , hence T(G) onto T{G').
(iii) The restriction map K0(ZG)->KQ(ZH) sends ([r, 2G]) to ([r, S J ) ,
hence T(G) onto T(H).
(iv) The exponent of T(G) divides the Artin exponent A(G). (For the
Artin exponent, see [6].)
(v) T(D(2n))=0(n>2), T{Q{2n))=Zj2 (n>3) and T(SZ)(2w))=Z/2(w>4),
where D(2n) [resp. Q{2\ SD(2)] is the dihedral [resp. quaternion, semi-dihedral]
group of order 2n. These groups are called the exceptional groups.
Theorem 1.3 ([10]).
( i ) If G is a non-cyclic p-group (p: an odd prime), then T(G) is the cyclic
group of order \G\jp.
(ii) If G is a non-cyclic and non-exceptional 2-group, then T(G) is the cyclic
group of order \ G \ /4.
Let Gp denote a/>-Sylow subgroup of G.
Corollary 1.4. If T(G) vanishes, then Gp is cyclic when p is odd and G2
is cyclic or dihedral.
2. The inclusion kG
torn Dieck and Petrie defined the finiteness obstraction map
(2.1) p: v(G, h~) -* ®KlZWH),
where WH=NH\H and NH is the normalizer of H in G. They proved that
the following sequence is exact.
(2.2) 0 -* v(G, h) -2 v(G, /T) & ®K,(ZWH).
256 I. NAGASAKI
We recall the map p. (For details, see [3].) For any element x of v(G,
h°°), there exist homotpoy representations X, Y and a G-map / : X-+Y such
that x=X— Y in v(Gy h°°) and dtgfH is prime to | G | for each subgroup H of
G. A function d^C(G) is defined by d(H)=degfH for any (H) and called the
invertible degree function of x. Conversely, any d^C(G) with (d(H)y \G\)
= 1 for any (H) is the invertible degree function of some x in v(G, h°°). The
finiteness obstraction map p is described as follows. The (iZ)-component pH(x)
<=K0(ZWH) of p(x) is equal to
(2.3) SWH{d{H))-
where K. is the subgroup of G such that K/H=K and aKtL are certain integers
and d is the invertible degree function of x.
Proof of Theorem A. For any r which is prime to | G |, we take the func-
tion d(=C(G) such that d(\)=r and d{H)=\ for (ff)4=(l). BY (2-3)> w e h a v e
p^x)=SG(r) and piy(^)=0 for (i/)=f=(l), where c^ denotes the element of v(G>
h°°) represented by d. Hence T(G)=0 if p=0. Conversely if T(G) vanishes,
then SK=0 for any subquotient group K of G by Theorem 1.2. Hence p=0
and so kG is an isomorphism.
Corollary 2.4. L*tf G k D(2tt), Q(2n) or SD(2n). Then v(G, h)=v(G, I).
Proof. In the case of D(2tt), we have proved it in [8]. In the cases of
Q{2n) and 5D(2n), v(G, I) is the subgroup of index 2 of v(G, h°°) ([8]). On the
other hand v(G, h) is a proper subgroup of v(Gy h°°) since T(G)=ZI2. Hence
v(G, h)=v(G, /).
REMARK 2.5. If G is nilpotent, then Dim V(G, l)=DimV(G, h°°) ([3])
and hence V(G, h)=V(Gy I) for the above groups.
Corollary 2.6. / / v(G, h°°) vanishes, then T(G) also vanishes.
3. The inclusion j G
Let G be an abelian group. Then v(G, I) and v(G, h°°) were computed
by Kawakubo [5] and torn Dieck-Petrie [3] respectively and it is known that
the following diagram is commutative.
v{G, I) > v(G,
(3.1) a
Jlu{GjH) c Uu(G/H)
Q-/M : cyclic
HOMOTOPY REPRESENTATION GROUPS 257
Here u(GIH)=(Z/\GIH\*)/±l.
Furthermore, torn Dieck and Petrie showed the following commutative
diagram.
(3.2)
Hence we obtain
Proposition 3.3. Let G be an abelian group. Then
( i ) v(G, h)^v(G, I)xN(G),
where N(G)=II Ker SG/H. (If G is cyclic, then we put N(G)=1.)
B
6-/B ; non-cyclic
(ii) v(G, h")/v(G, h)sz®T(GIH).
B
Proof. These are obtained from the exactness of the sequence (2.2) and
the fact that T(G/H)=0 if GjH is cyclic.
Corollary 3.4. Let G be an abelian group. Then
V(Gyh)^V(G,t)xN(G).
REMARK 3.5. For any finite group, one can show that
\v(G,h~)lv(G,h)\^\®T(WH)\.
From now we shall prove Theorem B. Theorem B is proved by the fol-
lowing lemmas.
Lemma 3.6. / / N(G)=1 for a non-cyclic abelian group G, then \ G \ = 2" • 3m
(ny m>0).
Proof. If a ^ >-Sylow subgroup Gp (p>5) is non-cyclic, then there exists
a subgroup L such that G\L is isomorphic to Zjp X Zjp. Since Ker SG/L is
non-trivial by Theorem 1.3, Gp must be cyclic. We may put G=G2xGzxC,
where C is a cyclic group with ( |C | , 6)=1. We prove that C is trivial. As-
sume that C is non-trivial. Since G is non-cyclic, there exists a subgroup K
such that GjK is isomorphic to ZjqxZjqxZjp (q=2 or 3, p>5). The Artin
exponent A(GjK) is equal to q and so T(GjK) is a 5-group by Theorem 1.2.
On the other hand, it is easily checked that the exponent of u(GjK) is not equal
to q. Hence Ker 5 /^^ -4=1 and so i\T(G)=}=l. This is a contradiction. There-
fore C is trivial.
258 I. NAGASAKI
Lemma 3.7. Put G=Zj2xZl2xZftm (m>l). Then Ker SG4=1 if m>2
and Ker SG=1 ifm=\.
Proof. Since the Artin exponent A(G)=2 and \u(G)\ =2-3IM"1, the Swan
subgroup T(G) is isomorphic to 1 or Z/2. Moreover r (Z/2xZ/2xZ/3)=Z/2
([+]> 17]). Hence T{G)=Zj2. Thus the desired result holds.
Lemma 3.8. Pstf G=Z/2"xZ/3xZ/3 (n>l). Tfew Ker SG*d *
Ker SG=l if n=l.
Proof. The proof is similar to the proof of Lemma 3.7. The details
are omitted.
Lemma 3.9. Let G2 be a non-cyclic abelian group of order 2n. We put
G=G2X Z/3. Then Ker SG2=l and Ker S c = l .
Proof. By Theorem 1.3, it is clear that Ker SG2=1. We consider the
restriction map
R = (resG2, res*): T(G) -> T(G2)®T(K),
where K is a subgroup which is isomorphic to Z\2 X Z\2 X Z/3. We show that
R is surjective. Take any element (a, b) in T(G2)(BT(K). Then there exists
reZwi th( r , |G|) = 1 such that resG2 SG(r)=SG2(r)=a. Vut c=resK SG(r)=SK(r).
If c=tb, then take (2W—l)r [resp. (2n+l)r] instead of r when n is odd [resp.
even]. Then
and
if
 n i s odd
The last equality follows from the facts that T{K)=Zf2, 5^(5)4=0 and 5^(7)
=t=0. Hence R is surjective.
The orders of T(G2) and T(K) are 2""2 and 2 respectively. Since \u(G)\
=2-\ Iu(G) I = I T(G) I. Hence Ker Se=l.
Lemma 3.10. Let G3 be a non-cyclic abelian group of order 3™. We put
G=Z/2xG 3 . Then Ker SG3=1 and Ker Se=l.
Proof. This follows from the comparison between the orders of u(G)
and T(G).
Lemma 3.11. We put G=G2xG3for the above G2 and G3. Then Ker SG
1
HOMOTOPY REPRESENTATION GROUPS 259
Proof. The restriction maps T(G)-> T(G2 X Z/3) and T(G)-*T(G3) are
surjective. Since | T(G2xZ/3)\ =2n~1 by Lemma 3.9 and \T(G3)\=3m^ by
Theorem 1.3, we have | T(G) \ >2n-^Zm'\ Hence Ker SG=1.
Proof of Theorem B. Assume that jG is an isomorphism (i.e. N(G)=1).
By Lemma 3.6, G=G2xG3. If both G2 and G3 are cyclic, then G is cyclic.
If G2 is cyclic and G3 is non-cyclic, then G2= 1 or Z/2 by Lemma 3.8. If G2
is non-cyclic and G3 cyclic, then G3=l or Z/3 by Lemma 3.7. If both G2 and
G3 are non-cyclic, then G=(Z/2)nx(ZI3)m by Lemmas 3.7 and 3.8. Converse-
ly, if G is one of the groups (i )-(vi), then N(G)==1 by Lemmas 3.7-3.11.
4. The finite groups G with v(G9 h°°)=0
In this Section we determine the finite groups with v(G> h°°)=0. We
first show the following result.
Proposition 4.1. Let C be a cyclic subgroup of G. Then the restriction
map
res: v(G, h°°) -> v(C, h°°)
is surjective.
Proof. Let d^C(C) be an invertible degree function representing
v(C, h°*). We can choose an integer aK such that d(K)-\-aK\C\ is prime to
\G\ for any subgroup K of C. Then d'(K)=d(K)-\-aK\C\ is also an invertible
degree function representing x. (See [3].) We define e^C(G) by
d'igHg'1) if (H) ^  0(G) with gHg~l c C
1 otherwise.
This is well-defined since C is cyclic. Let j> G z;(G, A°°) be the element represented
by e. Then resj>=# since J is an invertible degree function of resy.
In the abelian case, we have
Lemma 4.2. Let G be an abelian group* Then v(G> h°°)=0. If and
only if G is isomorphic to 1, Z/2, Z/3, Z/4, Z/6 or Z>(4) (=Z/2x Z/2).
Proof. Using the isomorphism v(G> h°°)^J[u(GIH)y one can easily see it.
By Lemmas 4.1 and 4.2, we have
Lemma 4.3. If v(G, h°°) vanishes, then any cyclic subgroup C of G is iso-
morphic to 1, Z/2, Z/3, Z/4 or Z/6.
On the other hand, if v(G9 h°°) vanishes, then the Swan subgroup T(G)
260 I. NAGASAKI
also vanishes (Corollary 2.6) and hence we have the following conclusion by
Lemma 4.3 and Corollary 1.4.
Lemma 4.4. Ifv(G, h°°) vanishes, then a 2-Sylozo subgroup G2 is isomorphic
to 1, Z\2, Z\\> D(A) or D(8) and a 3-Sylozo subgroup G3 is isomorphic to 1, or
Z\3 and a p-Sylozo subgroup Gp (p>5) is trivial.
We consider a non-abelian group G. Suppose that v(G, h°°) vanishes.
Then \G\=6, 8, 12 or 24 by Lemma 4.4. If | G | = 6 , then G is isomorphic
to D(6). If | G | = 8 , then G is isomorphic to D(8) by Lemma 4.4. If |G|
= 12, then G is isomorphic to A4, D(12) or Q(12). In the case \G\ =24, G2 is
isomorphic to D(S) by Lemma 4.4. From Burnside's book ([1] Chap. 9, 126.),
G is isomorphic to one of the groups: D(24), D(8)X-Z/3, S4 and K=(ay b, c\a*
=tf=cz=\, bc=cb, b~1ab=a-\ a"1ca=c"1y. However Z)(24) and D(8)xZ/3
are omitted by Lemma 4.3. Since K has a subgroup which is isomorphic to
Z/2xZI2xZ/3, the Swan subgroup T(K) is non-trivial and K is also omitted
by Corollary 2.6. Therefore, in the non-abelian case, if v(G, h°°) vanishes,
then G is isomorphic to one of the groups: Z)(6), Z>(8), D(12), 0(12), A4 and S4.
We proved the following formula in [8]. (See also [2].)
Proposition 4.4. For any finite group,
(4.5) Iv(G, h")I = 2"-1fl(G)*\n<p(\WH\),\n
zohere <p is the Euler function and n is the number of conjugacy classes of subgroups
ofG.
By computing \v(G, h°°)\ as in [8], one can see that \v(G, h°°)\=l for G
=D(6), Z>(8), Z>(12), A4 or S4 and \v(G, h~)\ =2 for G=Q{12). Therefore
we have
Theorem 4.6. v(G, h°°) vanishes if and only if G is one of the following
groups: Z\n («=1, 2, 3, 4, 6), D(2n) (n=2, 3, 4, 6), A4 and 54.
As a remark, there exist infinitely many groups with v(Gt \ ) = 0 (X=h or
/). Indeed we have
Proposition 4.7. Let G be an abelian group. Then v(Gy I) vanishes if
and only if G=(Z/2)nx(Z/4)m or (Z\2)nx (Z/3)"1 (w, m>0).
Proof. One can see it by using the isomorphism v(G, l)^TL u(G/H).
B
G/H: cyclic
By Proposition 4.7 and Theorem B, we have
Corollary 4.8. Let G be an abelian group. Then v(G, h) vanishes if and
HOMOTOPY REPRESENTATION GROUPS 261
only if v{G, I) vanishes.
References
[1] W. Burnside: Theory of groups of finite order, Cambridge University Press,
Cambridge, 1911.
[2] T. torn Dieck: The Picard group of the Burnside ring, Crelles J. Reine Angew.
Math. 361 (1985), 174-200.
[3] T. torn Dieck and T. Petrie: Homotopy representations of finite groups, Inst.
Hautes Etudes Sci. Publ. Math. 56 (1982), 129-169.
[4] S. Endo and Y. Hironaka: Finite groups with trivial class groups, J. Math. Soc.
Japan 31 (1979), 161-174.
[5] K. Kawakubo: Equivariant homotopy equivalence of group representations, J.
Math. Soc. Japan 32 (1980), 105-118.
[6] T.Y. Lam: Artin exponents of finite groups, J. Algebra 9 (1968), 94-119.
[7] T. Miyata and S. Endo: The Swan subgroup of the class group of a finite group,
to appear
[8] I. Nagasaki: Homotopy representations and spheres of representations, Osaka J.
Math. 22 (1985), 895-905.
[9] R.G. Swan: Periodic resolutions for finite groups, Ann. of Math. 72 (1960), 267-
291.
[10] M.J. Taylor: Locally free class groups of groups of prime power order, J. Algebra
50 (1978), 463-487.
[11] S.T. Ullom: Nontrivial lower bounds for class groups of integral group rings,
Illinois J. Math. 20 (1975), 311-330.
Department of Mathematics
Osaka University
Toyonaka, Osaka 560
Japan



Osaka University Knowledge Archive

http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-02402-2.pdf

Homotopy representation groups and Swan subgroups

Abstract

Similar works

Full text

Available Versions

Institutional Repositories DataBase (IRDB)

Osaka City University Repository

Osaka University Knowledge Archive