Math. Ann. 273, 45-64 (1985) Mathematische Annalen �9 Springer-Verlag 1985

Logarithmic Descriptions of K'I( pG ) and Classgroups of Symmetric Groups Robert Oliver Matematisk Institut, Aarhus Universitet, Ny Munkegade, Bygning 530, DK-8000 Aarhus, Denmark

For a finite group G, and any prime p, set

K i (Jl!pG) = K 1 (~pG)/ S K , (~I,G) ~ Im [K x(~pG) ~ K 1 (QpG)].

Our main result here is a logarithmic description of K'l(~pG)(p), analogous to the descriptions in the p-group case given, for example, in [-8, Theorem 2] and 1-11. Proposition 2.3].

More precisely, a homomorphism

Fo " K'~ (J~pG)~ Ho( G ; ~pG) ,

where G acts on ~pG via conjugation, is defined by setting

Fa(u ) = log(u)-- 1 q~(log(u)) (q~(E aigi) = ~, aig~) . P

Let Gr ~ G denote the subset of p-regular elements (i.e., those g ~ G of order prime to p); and let G act on ~p(G,) by conjugation. In Sect. 1, we will show:

Theorem 1.8. I f p is odd, then there is a short exact sequence

O~K'~(~pG)~p) ~r '~') , Ho(G; ~pG)OH~(G; ~p(G~)) ~ '~-~) , H~(G;~p(G,))~O ;

where in particular, cI)(~ gi@aihi) = ~_, gi@aihl~ for gi ~ G, hi ~ Gr, and a i ~ ~p.

An immediate consequence of this is:

Corollary 1.9. Let ~" HI(G; ~p(Gr))--*HI(G; ~p(Gr)) be as above and set

H~(G; ~p(Gr))~ = Ker(~ - 1); H~(G; ~p(G,))~ = Coker(q~- 1).

Then, if p is odd

Ker (Fa) = tors K'~ (~pG)(p) ~- H I ( G ; ~p( G,) ) ~ ,

and

Coker(Fo) ~ H, (G; ~p(G,))..

46 R. Oliver

Theorem 1.8 and Corollary 1.9 are, in fact, stated and proven below for arbitrary p; but extra terms are necessary when p = 2. Note that if G is a p-group, then G, = { 1 }, �9 = 1, and so

Ha(G; ~p(Gr)) = Ha(G; ~p(Gr)) '~ = HI(G; ~p(Gr))a, ~ G ab.

These sequences, first announced in [12], are frequently useful when working with Whitehead groups or projective class groups of finite groups not of prime power order. For example, they can be used to simplify considerably the analysis of the Whitehead transfer homomorphism for Sa-bundles, as carried out in [13, Sect. 4]. In this paper, we use them to derive a formula for the order of the projective class group of a symmetric group.

Recall [1, Sect. 28] that for any n, the group ring Q[Sn] is a product of matrix rings over Q. In particular, any maximal order OJ~ ~ KS, is a product of matrix rings over 7Z, and so

D(ZS,) = Ker [/s =/(o(ZS,) .

Corollary 1.9, together with Mayer-Victoris exact sequences, are used to derive a formula for [D(ZS,)[ (Theorem 2.5), in terms of the partition function p(n). These techniques can also be used to get information about the exponent of D(ZS,), by comparing it with the Artin exponent (see the remarks at the end of Sect. 2). In principle, they can be used to get much more information about D(ZS,); but seem likely only to lead to a description which (at best) is equivalent to that shown by Taylor in [17].

Throughout the paper, for n > 1, C, denotes a (multiplicative) cyclic group of order n, and ~, a primitive n th root of unity. If K is a field, then K#, denotes the smallest field containing K and the n th roots of unity (i.e., not K| If ZGC=?gi~C=QG, where ~)l is a maximal order, then we identify when convenient K'I(~pG) and ' ^ K l(gJlp) as subgroups of K a(~pG). And for any ring R, J(R) denotes the Jacobson radical of R.

1

Throughout this section, we fix a prime p, a finite unramified extension K ==~p, and let AZ=K and q~=~o~eGal(K/t~p) be the ring of integers and the Frobenius automorphism. The goal is to construct a short exact sequence giving an additive description of K](AG)tp ), for any finite group G. A homomorphism from K'~(AG) to Ho(G; AG) must first be constructed, where G acts on AG via conjugation.

Theorem 1.1. Fix a finite group G, and let J = J(AG) be the Jacobson radical. Then there is a homomorphism

log : Ka(AG)tp)~ K~(AG, J)~Ho(G; K G)

such that for any l + M ~ GL(AG, J):

1 2 1 3 log([1 + M]) = trace(logM) = t r a c e ( M - ~ M + x M - ...) e Ho(G; KG).

Proof. Note first that logM converges for M ~ M~,(J): J is nilpotent in A/pk[G] for any k > l [2, Proposition 5.15], and so M"/n~O as n ~ .

Logarithmic Descriptions of K'I(~pG ) 47

Assume we can show that the composite

trace o log : GL(AG, J)--*MOO(KG)~Ho(G, KG) (1)

is a homomorphism. For any M E Moo(J) and R e GL(AG),

trace o log(1 + RMR) = trace(R �9 log(1 + M). R- 1) = trace o log(1 + M) ;

and so (2) factors through a homomorphism

log : K 1(AG, J) = GL(AG, J)/[GL(AG), GL(AG, J)] ~H0(G; KG).

Since AG/J is a product of matrix rings over fields of p-power order, Kz(AG/J ) = 0 and PXIKI(AG/J)I < oo. By [20, Theorem 10], KI(AG, J) is a pro-p- group, and so KI(AG, J)~K~(AG)(p).

It remains to show that (1) is a homomorphism. For each n > 1, let W, be the set of formal (ordered) monomials of length n in two variables x, y. For w e W,, set

C(w) = orbit of w in IV, under cyclic permutations,

k(w) = #(occurrences of xy in w), and

r(w) =(coeff. of w in log(1 +x+y+xy ) )

= ~2(--i=0k'w) 1)._i+1 1 (k(w))__n_i "

for each i, w can be written in (k(W)) ways as a product of i (xy)'s and n - 2i x's or

y's. Fix elements M, NeMoo(J). For any n> 1, we W,, and w'eC(w),

trace (w'(m, N)) = trace(w(M, N)) e Ho(G; KG) :

since trace(RS) = trace(SR) for any R, S e MOO(KG). Thus,

- Z Y, Z trace~ log(1 + M + N + MN) - =1 w~ w,/c [w,~cw r(w')] trace(W(M, N)). (2)

For fixed w E W,, if ICwl = n/a (i.e., w has cyclic symmetry of order a), and if

k=max{k(w3:w' e Cw},

then Cw contains k/a elements with k - 1 (xy)'s (i.e., those of the form y. . . x) and (tl-k)/a elements with k (xy)'s. So

Y~ r(w')= 1 ~ ( _ l ) , _ i + ~ l [ (ki) ( ~ ) 1 w'~c~ ai=o n - i (n -k ) +k k 1

a i = o

I ~ if ( - 1 ) "+11- if k=O(soa=n) . n

48 R. Oliver

Formula (2) now takes the form

traceolog((1 + M ) ( I + N ) ) = ~ ( - 1 ) " + l - t r a c e + - n = l

= trace o log(1 + M) + trace o log(1 + N) ;

and hence (1) is a homomorphism. []

Now define, for any finite G,

F = FAG = 1 -- P �9 o log : KI(AG)(p)--~Ho(G; KG) ;

where ~(~aigi)=Zq~(ai)gf for ai~K, gi eG (recall that ~o is the Frobenius automorphism of K). As will be seen later, FAG has image in Ho(G; AG), and is the main ingredient in the exact sequences involving K~ (AG) described above. We first consider the p-group case; the next theorem is basically a reformulation of earlier results.

Theorem 1.2. Fix a prime p, an unramified extension K/(~p with rin9 of inteoers A c_ K, and a p-group G. Define

(co, O)= (COAG, O A6) : K'I(AG)(p) ~ ( G"b| A)O(A/2) ,

and (v, O) = (rAG, OAG ) : Ho(G; AG)~(G"b| ,

by settino, for gi ~ G and a, a i 6 A (and ~, ai ~ A/2 their reductions) :

(co, 0) ((1 + pa) (1 + Z a,(gi- 1))) = (Z g~| d) ;

(v, O) (E aigi) = (Z 9i| Z ai) .

Then co, O, and F are all well defined on K'I(AG)(p), Im(F)C=Ho(G; AG), and the sequence

, (r,o,~O) _ . _

O-~KI(AG)(p) ~ Ho(G; AG)G(G"b|

('o 1| 1) ~~ , (G"b| ~O (1)

is exact.

Proof. Let I = Ker [e : AG-~A] be the augmentation ideal. Since e is split, there is a natural decomposition

KI(AG)(m~-(1 + pA) x K,(AG, I).

This makes it clear that the above formulas induce homomorphisms co, 0 well defined on K~(AG)(p) at least: co is defined via projection to K~(AG, I) and 0 via projection to 1 +pA.

That Im(F)_C_Ho(G;AG) was shown in [8, Proposition I0]; and F(SK~(AG))=O since SK~(AG) is finite [and Ho(G;AG ) is torsion free]. By naturality, co and ~q factor through KI(A[G"b])(p); and so (co,~)(SKI(AG))=0 [since SKI(A[G"b])=O]. Thus, F, co, and 0 are all well defined on K'1(AG)(p).

Logarithmic Descriptions of K'x(~pG)

Sequence (1) splits as a sum of sequences

O~K'~(AG, I) (r.~,)_, Ho(G,I(AG))O(G~b|

and

49

( v , l | ,Gab| (2)

(F , ~o0) (0, ,p - 1 ) 0~1 +pA -----~ A | , A/2~O. (3)

By [11, Proposition 2.3] (the equivalence of the two definitions of v), for any x ~ I(A~):

(1| -- (p)) (~o(1 +x))=vF(1 +x).

In other words, the composite in (2) is zero. Also, the sequence

0 ~ v i.c~ A 1-~o A Xr) ~p"~0

is exact: q~ generates Gal(K/C)v), K/C)p is unramified, and so A is projective over ~;[Gal] (and hence cohomologically trivial) by [5, Theorem 9.1.2]. So

Ker(4, - 1) ~ Coker(4~ - 1)2 G "b ,

and (2) is exact if and only if the sequence

inc l t F ( t r | O~G ab , Ka(AG, I)---~ Ho(G,I(AG)) ~G"b--,O

is exact. This follows from [8, Theorem 2]: note that

ab I Wh'(AG)~-Ki(AG)/(A * x G )~--KdAG, I)/G ab.

If p is odd, then

r ( l + p A ) = (1 P P .

Since 1 +pA is torsion free, F is an isomorphism in this case, and (3) is exact (A/2 = 0). If p = 2, then

f ( l +4A)=(1 -'q~)(log(1 +4A))=(1- �89 O(l + 4 A ) = 0 .

]:'or any xe A, q)(x)-xZ(mod2A), and so

F(1 +2x)=(2x-2x2+ . . .)-(~o(x)-~o(x2)+ . . . ) - -q)(x)+q~Z(x) (mod2A).

Since ~00(1 + 2x)= ~o(x), this shows that

(F, ~o0) (1 + 2A) = {(x - (p(x), x) e A x A/2} = Ker(0, q~ - 1)

(O:A--.A/2 is in this case just reduction mod2). Also, K e r ( F ) = { + l } and ~p()( - 1) = l, so that (3) is exact in this case also. Thus, (1) is exact, and the theorem is proved. []

It is not hard to generalize Theorem 1.2 to the case of p-elementary G - i.e., G ~ C, x 7z, where p,~n and 7t is a p-group. To get from these to arbitrary finite G, we will need some of the induction techniques of Dress [3-1. The basic idea is that of "computability" with respect to certain subgroups.

50 R. Oliver

Let F be a functor from finite groups and monomorphisms to abelian groups sending inner automorphisms to the identity, and let cg be a class of finite groups such that H ccg if H_c G ccg. Then F is called Cg-computable if for any finite G, the natural homomorphism

~ { F ( H ) : H C G, H ~ cg}__.F(G)

is an isomorphism; where the limit is taken with respect to inclusion and conjugation homomorphisms between the subgroups. Of particular interest here will be the pg-computable functions; where pg denotes the class of p-elementary groups.

For any finite G, G, will denote the set of p-regular elements in G: the elements of order prime to p. We let A(Gr) denote the A-module with basis G, with the G-action induced by conjugation.

Theorem l.3. The functors KI(AG)~v), K'I(AG)~p), SKI(AG), H,(G;AG), and H,(G; A(G~)) (n >0) are all pg-cornputable as functors of G.

Proof. Step 1. Fix an unramified extension L= K, and let B C_=L be the ring of integers. Let t" n-~Gal(L/K) be a homomorphism, where n is a p-group, and set

=Ker(t). Let Bn r denote the corresponding twisted group ring:

B1t'= • Bg; (blgO'(b2o2)=(bl "t(gO(b2))(glg2) (bi~B, gicrO. g e t :

We claim that

K 1 (B~t)r ~- Ho(~/q; K'~(Bq)~p)), K l(BTzt)tp) ~- Ho0z/Q; K ~(BQ)r , (l)

SK ~(BTt t) ~ Ho(n/O ; SK,(Bo)) ;

where all isomorphisms are induced by the inclusion BoCB~ t. For SK~, this is shown in [9, Proposition 7]; and the other two cases are shown in the course of proving that proposition. However, since the proof for K't (Bid) is simple, and is the case needed later in this paper, we repeat that proof here.

By [9, Lemma 5], K'1(BQ) with the 7r/q-action induced by

(t, conj) : n/o-~Gal(B/A) x Out(Q)

is cohomological trivial. Consider the induced maps

i n d t r f

KI(Bo)~p ) , , t " Kl(Bn )iv) ' K I ( B ~ ) ( p )

The composite is induced by tensoring with B~ t as a Br and hence is the norm map for the ~/Q-action. Since K'~(BQ) is cohomologically trivial,

! , t Ker(ind)_C_ Ker(norm) = Ker [ K l ( Bo)tp)--~ HoOr/o , K I ( BQ)~v)) ] c_ Ker(ind);

and so the indusions are equalities. By [9, Lemma 3], ind is onto, and hence / t t induces an isomorphism of Ho(n/O; Kt(Br ~) to KI(BTr )tpr

Step 2. We now prove the theorem for K'I(AG)tp); the proofs for KI(AG)~p} and SK~(AG) are similar (SKI(AG) is a p-group by [21, Theorem 2.5]). For any finite G, set

F(G)=K~(AG)r F(G)=lim{F(H) U=G,H~p ,~} .

Logarithmic Descriptions of K'I(~,pG) 51

Let Ia : f (G) - ,F(G) be the induced map, and set

F'(G) = Ker (I~), F"(G) = Coker (I~).

All of these - F,/7, F', and F" - are functors on the category of finite groups with homomorphisms.

Let pK8 denote the class of p-K-elementary groups: G ~ pKg if G ~ C, x ~, where p,fn, ~ is a p-group, and such that if K[C,] ~HL~ is the decomposition as a product of fields, then the conjugation action of ~r leaves the factors invariant. By [3, Propositions 1.1' and 1.2] (and [-9, Proposition 1]), F is pKg-computable. Thus, if F is not p~-computable, then there exists G ~ C, x ~ ~ pKg (pXn, ~ a p-group) such that IG is not an isomorphism, but I n is for all H~ G.

Let Q be the set of primes dividing n. For each S__ZQ and each H__z G, let e~ e End(H) be the idempotent which is the identity on some p-Sylow subgroup of H and on all q-Sylow subgroups for q~S; and which is trivial on q-Sylow subgroups for q ~ Q\S. The es n are unique up to inner automorphism, and induce decompositions

F(H)=s@_QFs(H), ff(H):s~cQffs(H), = �9 =

(and similarly for F ' = OF}, F " = | Here, for SC__Q,

Im [(e~1). "F(H)~ F(H)] = r@_~ s FT(H); (2)

and similarly for the other splittings. The Fs, ffs, etc., are all functors on the category of subgroups of G with inclusions and conjugation. By the naturality of the splittings,

ffs(G) ~ lira {Fs(H ) �9 H c G, H C p~}

for all S; and the following sequence is exact:

O~F,s(G)~ffs(G ) ~ ' ~ Fs(G)~F~(G)~O.

By assumption, F'(H)= F"(H)= 0 for all H ~ G. In particular, by (2), Us(G ) =F~(G)=0 for all S~Q. So to finish the proof that IG is an isomorphism, it remains only to show that (IG)Q is one.

Set

oW= { H C= G : H e pWo ; qIIH [ for all q e Q } .

By construction, FQ(H) = 0 for any H__c G not in W. Furthermore, o~ contains a unique maximal element Go = C, x Q, where

= Ker [~z~Aut(C,)]

(recall that G = C, x ~). It follows that

ffQ(G) _ ~ {FQ(H) : H __c G, H ~ pg} - lim~FQ(H) ~ Ho(g/r FQ(Go)) ; I I ~ , , ' f

where ~/r ~ G/Go acts on FQ(Go) via conjugation.

52 R. Oliver

Let ti = r [ q, the product of the distinct primes dividing n. Write q~Q

K[C,]_~ r[ K | • f i Li ; A[C,] ~- [I A| x f i B i ; ~ga ln i = 1 ~/(d[n i = 1

where the L i are finite unramified extensions ofK 1 and B i _c Li is the ring of integers (p~Vn, so A[C,,] is a maximal order). Then we can identify

Fe(Go) ~ f i K'~(Bi[Q])to); FQ(G) ~ f l K'~(B,[~]')~,). i = 1 i = 1

So by (1) (Step 1),

fro(G) ~- Ho(~/q; FQ(Go) ) _~ FQ(G) .

This finishes the proof that (IG) e (and hence IG) is an isomorphism. By induction now, F is pg-computable.

Step& It remains only to show that H,(G;AG) and H,(G;A(Gr) ) are pg-computable for all n> O. If 91 . . . . , gk are conjugacy class representatives for elements in G, then

k k

H.(G; AG) ~- • H,(G; Ind~,)(A(gz)))_~ ~, H,(Z~(g~))| i = 1 i = l

In the notation of Dress, H, is a Green module over H~ and hence (p-locally) computable with respect to p-subgroups. It follows that H,(G; AG) is computable with respect to subgroups H x (gi> for p-groups Hc=Z~(g~); i.e., with respect to p-elementary subgroups of G. The proof for H.(G; A(Gr) ) is similar. []

Three lemmas are still needed before the main theorem can be proven.

Lemma 1.4. Fix n > l such that p~n; and choose an isomorphism k

f : K[C,] ' I-I Li, where the L i are fields. Then Li/(~ p is unramified for all i. Let i = l

B i l L i be the rin 9 of integers, and let (p=Gal(K/(~p), 99ieGal(Li/~p) be the Frobenius automorphisms. Then q~(()=~P(~01(~)=(P) for any root of unity ( in A*(B*) of order prime to p; and the following diagram commutes:

K[C,] , K[C.] [ [

f

h L i m ~ , i = 1 i = 1

Proof. Note first that A/p[CJ~-I-IB]p is a product of fields: p~n, and K/@p is unramified by assumption. So pBi c= B, is the maximal ideal for each i, and Li/~p is unramified.

Let #____ A*, #~ e (Bi)* be the groups of roots of unity of order prime to p. Then l~-~ (A/pA)*, ~o(/~)=/~, and ~o(a)- aP(modpA) for all a e A. It follows that q~(~)= ~P for all ( e #; and similarly for (e/~. Finally, since f ( g ) e [ I #~ for all g e C,,

(I-I rpl) (f(9)) =f(9) p = f ( g p) =f(q~(9)) ;

and so (1) commutes. D

Logarithmic Descriptions of K'I(~pG) 53

For any finite G, and any g E G of order n, define

N~(g) = {x ~ G; x g x - 1 = ga, some a ~ Gal (K~, /K) c= (Z/n)*}.

Two elements g, h ~ G are called K-conjugate if g is conjugate to h a for some a ~ G a l ( K ( , / K ) (n=lh[). Of course, N~p(G) and t~p-conjugacy are defined analogously.

Lemma 1.5. Fix a finite group G and a p-regular element g ~ G, and set n = Ig[. Then (i) N~(g) = {x ~ G : xgx 1 = x p,, [K : ~p-][i} and

N~op(g) = {x ~ G" x g x - ~ = x p', i 6 Z} .

(ii) I f e = m i n { i > O : g p' conjugate to g}, then

#e { K-conjugacy classes in g's @p-conjugacy class}

= ([K :~p], e) = [Fix(ep e, K) : I~p].

Proof Set d = [K: ~p]. The extensions K#. /K and ~p~./~p are unramified, and so

((p : (-~(P) e Gal(~)p(./~)p), (q~d: ~_~(v d) e G a l ( K ( . / K )

are generators (see Lemma 1.4). So for any h 6 G, h is (~p-(K-)conjugate to g if and only ifh is conjugate to gV' for some i ~ 7Z (i ~ d7Z). So (i) is immediate. Also, from the definition of e, g is K-conjugate to gP' (any i) if and only if

i e dZ + eZ = (d, e)7Z.

Thus, {g, gp, gp2, .... gP~"'"~-'} is a set of K-conjugacy class representatives for elements in g's t~p-conjugacy class. That (d, e) is the degree of Fix(C, K) over t~p is clear. C3

The next lemma is more technical. For any G, let

q~ : Hi(G; A(Gr))~HI(G; A(Gr)),

~b : Ho( G ; A/2( G,) ) ~ Ho( G ; A/2( G,) )

be the homomorphisms induced by ~:Zaih~+~.tp(a~)h f on coefficients (~0 e Gal (K/l~p) is still the Frobenius automorphism). For example, for g e G, a e A, and he Gr, tb(g|174 p. Let

H~(G; A(G,)) ~ Ho(G; A/2(G,)) ~

denote the subgroups of elements fixed by 4~.

Lemma 1.6. For any f inite G,

[tots K'~ (AG)(p)[ = ]H~(G; A(Gr))~I �9 IHo(G; A/2(G,))~{.

Proof Let gl . . . . , gk be ll)p-COnjugacy class representatives for elements in G~. For each i, set

e~ = min {s > 0 : gf~ conjugate to gi}

K i = Fix(tp"', K), Ai = F ix( r A)

ri= ~ {K-conjugacy classes in g[s ~p-conj. class} = [Ki: d)p]

54 R. Oliver

N i = N~(gi) = { x ~ G" xg lx - 1 = g~,,, some s E [K" ~p]Z}

IVi = N~"(gi) = {x ~ G : xg ix - 1 = geS, some s e 2~}

z , =

(see Lemma 1.5). Then

,=1 l~I Fix {/'e , ~'~11A/2(g~S)). [Ho(G; A/2( G~) )~I

k k = I-I IFix(~o~, Z/2)l = ~ IAJ2Ail (1)

i = 1 i = 1

k k

= 1-[ (2,P) ~'= 17 ]tors(A*)(p)l r i = 1 i = 1

[Since K / ~ , is unramified, the only possible torsion in (K*)~p) is + 1.] For each i, let Ni act on Z'] b via (x, g)--*x- lgx; and on K by setting: x(a) = ~p~(a)

pS if xgix-1 = 9i �9 By construction, K i is the fixed field of this action, and Gal(K/Ki) ~ - R J N i. Since K / K i is unramified, A is a free Ai[Ni /Ni]-module of rank 1:

(A/pA) ~- (Ai/pAi) [Ni/N~]

by the normal basis theorem [19, Sect. 67], and any isomorphism lifts to an isomorphism on A. It follows that

k e l - - 1

IH,(6; A(Gr))*I: vix( , ~ A(g 3)

k k

= I-I [Fix(~,ZTb| = 1-I IFix(~7,, ZT~| i = 1 i = 1

k k

-- F[ IFix(Ni, Z~b| = [ I IH~ "~7"ba,(p).V'. (2) i = 1 i = 1

On the other hand, by [9, Theorem 2], l

t o r sK ' l (AG)(p~ ~, H~ ; (Za(hi))ab)tp)| i = l

where hi . . . . ,ht is any set of K-conjugacy class representatives for p-regular elements in G. The lemma now follows from this together with (1) and (2) (note that l= Z rl, by definition). []

The exact sequence for describing K'I(AG)tp) can now be constructed. For any finite G, and any g ~ G, let gr e (g) denote the p-regular part; specified uniquely by the conditions: g = gsgr, [gs, g,] = 1, p/]g~[, and [gs[ = f f (some i). Define

VA G : Ho(G; A G ) ~ H I ( G ; A(Gr)) ; Oac,:Ho(G; AG)-*Ho(G; A/2(Gr))

by setting, for age A and g~ ~ G (1 < i < k):

v a i g i = Y~, g i | ; 0 a ig i = a i (g i ) r (di ~ A/2) . i i = 1 i i =

As in Lemma 1.6, q~ denotes the endomorphisms of H1(G;A(Gr) ) and Ho(G; A/2(G,)) induced by (Z, aig~--'Z r on the coefficients.

Logarithmic Descriptions of K'~(ZpG) 55

Theorem 1.7. Fix a prime p, an unramified extension K/~p, and a finite group G; and let A ~ K be the ring of integers. Then

(i) There are unique homomorphisms

=tOAC , "K](AG)(p)~H~(G; A(G,)), O= Oa6 "K'~(AG)(p)~Ho(G; A/2(Gr))

which are natural with respect to group homomorphisms; and such that if G = C, • rt is p-elementary (pXn, rc a p-group), if

k u= ~ aiYi ~ 1 + J(AG) (gi ~ rt, a i ~ A[C,], J = J a c . radical),

i=1

and if a = ~ a i ~ 1 +pA[C,], then

o ( [ u ] ) = ~ gi| laieHI(G;A(G~)); O([u])= l-(a-1)eA/2[C,] . (1) i = 1 p

(ii) C(K~(AG)~p)=Ho(G, AG), where C=FAG= 1-- p ~ olog.

(iii) The sequence

(r, ~, ~0) O~K'~(AG)~p~- - - - . Ho(G; AG)OHa(G; A(G~))| A/2(G~))

v %1 ol) (o ,, Ha(G; A(Gr))OHo(G; A/2(G,.))~O (2)

is exact.

Proof. Assume first that G is p-elementary: G = C, • z, where p~n and ~ is a p-group. Identify

k k k

K[C,]~- I-] L i, A[C,]~- ~I B i, AG~- [ I B/[rt] ; i= l i=1 i=1

where the Ki/K are unramified extensions, and Bi ~ Li is the ring of integers (A[C.] is a maximal order, since p.~n). Then

O)AG = I ~ O)B,[r~] and GAG = l'-I 0B,[;'t]:

these clearly satisfy the formulas in (1) above. Also, by Lemma 1.4, the homomor- phisms FAG, ~A~, Oa~, and VA~ split as sums of the corresponding maps for the /3i[rt]. So the exactness of sequence (2) for G follows from the p-group case (Theorem 1.2).

Now let G be an arbitrary finite group, and let E denote the set of p-elementary subgroups. By Theorem 1.3,

' ~ i ' �9 Ka(AG)~p) = ~ K ~ ( A H ) , H0(G; AG) =~H~ AH), etc;

where the limits are taken with respect to inclusion and conjugation, as usual. Define

(~OA~, 0AG ) = lim.(O~an, dan ) �9 K'~(AG)~p~--*H~(G; A(G~))OHo(G; A/2(G~)).

56 R. Oliver

Since direct limits over E are right exact, the sequence

(r, ~o, ~0) K'~(AG)~p) , H0(G; AG)OH~(G; A(G,))OHo(G; A/2(G,))

~ H I ( G ; A( G,))O Ho( G ; A/2( G,) )~O is exact.

Consider the following diagram:

(3)

By the exactness of(3), square (4) commutes, and induces an isomorphism/~ and an epimorphism e. In particular, Coker(F) is finite. Since Ho(G; AG) and K'~(AG)(p) are ~p-modules of the same rank [21, Proposition 1.6], and Ho(G;AG) is torsion free, we get that

Ker(F) = tors K] (AG)~p).

By Lemma 1.6, Ker(F) and Ker(1-4~) have the same order, so e is an isomorphism, and sequence (5) is short exact. []

The last step in the proof of Theorem 1.7 - showing that (F, co, 40) is injective - can also be carried out by showing Hi(G; A(Gr)) and Ho(G; A/2(Gr)) to be "pd ~ acyclic" in the sense of Dress [3, Sect. 1]. This would imply that the derived functor lim 1, for the last term is zero.

One can define restriction maps on the Ho(G; AG) which commute with T and the transfer maps for K'I(AG). There are none, however, on HI(G;A(G,.)) or Ho(G; A/2(G~)) which commute with w or ~d and the transfer; and in fact there seems to be no restriction maps on the middle term in the exact sequence of Theorem 1.7 which makes (F, co, ~0) natural with respect to transfer. If there were, the last step of the proof of Theorem 1.7 could be simplified by using inverse limits. Note also that for p-K-elementary G, coAG and OAS need not decompose as direct sums according to the decomposition of AG as a product of twisted group rings.

The following corollary to Theorem 1.7 is immediate:

Corollary 1.8. Fix a prime p, an unramified extension K /~p with rin 9 of integers A c_ K, and a finite group G. Set

Hi(G; A(Gr)) ~ : {x e HI(G; A(Gr) ) : ~(X) : X} ,

Hi(G; A(G,))~ = H~(G; A(G,))/Im(1 - 4) ;

and similarly for Ho(G; A/2( G,) ). Then there is a short exact sequence, natural in G:

0 o H , ( G ; A( G,) )* @ Ho( G ; A/2( G,) )*---, K', ( A G)~p)-~Y~ Ho(G; A G)

--,H~(G ; A(G,))~@Ho(G ; A/2(Gr)), ~O.

In particular, IKer(F)} = [torsK'l(AG)r =lCoker(F)}. []

(4)

0 , Ker(F) , K'~(AG)~p) - - , Ho(G; AG) - , Coker(F) - -~ 0

Hi(G; A(Gr)) 1 ~, HI(G; A(Gr)) 0 ~ Ker(1 - ~b) --, @no(G; A/2(G~)) ' | A/2(G,.)) --" Coker(1 - 4) ~ 0

Logarithmic Descriptions of K'~(ZpG) 57

2

We now concentrate on the case of symmetric groups, deriving formulas for ID(ZS,)I and exp(D(2~S,)). The idea here is to compare D(ZS, ) with the "Artin cokernel" of S,:

A~(S,) = R~(S,)/32 {IndGn(R~(H)) : H G S,, H cyclic}.

The advantage of doing this is that the order and exponent of A~(S.) are already known.

Recall [1, Sect. 28, p. 196] that for any n, ff)S, is a product of matrix algebras o v e r ([~.

Throughout this section, p and q always denote primes.

Lemma 2.1. Fix some n > 2 and a prime p < n, and write n! = ptm where p Xm. Define

F,.p = T,.pO Fs, : K'~(~,S,)~p,~Ho(S,; ~pS,) ;

where Fs. is the homomorphism defined in Sect. 1, and

m y~l~(d) g d ~"'P(g)= qo(m) dl,. d

for g ~ S.. Fix a maximal order ?0l 9= ZSn, let 1/1 . . . . . V k be the distinct irreducible r and define

by setting, for any u ~ 1 + d(?Olp),

k

b.,p(u)= p - 1 y. logdet~(U,~p| p i = l

Then the following diagram commutes:

where a(g)= Ind~>([~]) for any g ~ S., and is an isomorphism.

Proof. It will suffice to show that (1) commutes on units exp(p2g)~(~+S,) * for .q ~ S. [these generate a subgroup of K] (~pS.) whose closure has finite index]. For 1 <<-i<_k, set P/=~p| for short. For any g ~ S , ,

k 6,,p(exp(p2g)) = p-- 1 y, log det~p(exp(p2g), l~i ) �9 [V/]

p i=1

k k _ p - 1 y, pZTr~(g ' Vii). [V~] = p - 1 ~ p2Zv,(g). [V/].

p i = l p i=z

58 R. Oliver

On the other hand, if H = (9) , then

=p2" mq)(m) i=LEd~l ! ~(~ )dim~H~ Vi)l ' [ VJ

(since Endes,(V~) ~tI~, all i)

m L F~/~(d) ,. 1 ~p2. ~o(m) ,=2., Ld~ d mm~ Hom~n(ll)(H/ga), V~IH )= �9 [Vii]

It remains only to show, for each ~H-module V, that

~~ ')Zv(9) = dZl,! P(~) dim~ Hom~H(ll)(H/gd), V) . (2)

It suffices to prove (2) when V is irreducible. Let s be the order of the action of,q on V; i.e., V-~(l~((s). Then sllgltn!; and so

~(d) dim~Hom~n(ll~(H/gn), V ) - 2 l~( d) din! (l --sldln! d -(p(S)

=q~(s)'#(s~)" l-I { l -- ~ : q prime' qln''q~s

"W-'L n! / s J- ~o(n!) T ~ , q)(n!) ~,

- ~ . �9 r~r zvtg).

This proves (2), and hence the commutativity of (1). Finally, a is onto by the Artin induction theorem [2, Theorem 15.4], and hence

an isomorphism by dimension count. []

Lemma 2.1 can now be used to study KI(!fJlp)/K'I(J~pS,).

Proposition 2.2. Fix n > 2, and let ~ ~= 7ZS, be a maximal order. Then

1-[ IK ,(@p/K'~(gs = IA~(S,)I. p<n Proof. The proof will be carried out by comparing ~,p-lattices in I~p@Rct(Sn) and Ho(S. ;~ ,S, ) . For convenience, if L1 and L2 are two such lattices, we write

[L1 : L2] = ILl : Llc~L2]/[L2 : LlmL2] (=pi, some i s 2~).

For each p<n, F,,p, ft,,p, ~,,p, and a denote the homomorphisms defined in Lemma 2.1.

Step 1. By construction, for any p < n,

[~,| if p is odd 6,,p(Kl(9)/p))= [~2 | if p = 2 .

Logarithmic Descriptions of K'I(C/pG) 59

Since R~(S.) is a product of matrix algebras over Q, 93/is a product of matrix ,algebras over Z, and so (whether p = 2 or p>2) :

[torsKl(gJ/p) I = [~e| (p- 1)rkR~ (s '0 . (1)

Also, by Corollary 1.8,

,torsK'~(~pS.)l=[Coker(Fs.)," torsK'l(~vS.)[lp]

=[rrT~.,p(Ho(S.;~pS.))'aF.,p(K'~(~pS.))]. torsK'~(~pS.)[~] (2)

(recall that 7~.,p and rr are injective).

Step 2. By definition,

A~(S.) = R~(S.)/~. {Ind~"(R~(H)) H c S., H cyclic}

= R~(S.)/(IndS"([Q]) �9 H c S. cyclic) = R~(S.)/a(Ho(S .; ~pS.)).

Thus, for any p<n, Lemma 2.1 applies to show:

]tors K l(~lp)[ [K l(~lp)/K'l(~pS,,)l = [Im(5.,p) " Im(crF.,p)] . l~ors Kt(~pS,,) I

=[~v| ] .prkK~Cs.)/torsK',(~.pS.) [ ~ ]

[by (1), (2)]

= [A~(S.)(r)]. [detG(7~ p)](p).prkR~(S.)/torsK](~pS.) [~]. (3)

I lere, for any k ~ Q, we write k(p) for the pth power part of k: i.e., k(p)= pi for some i ~ Z, and (k/kw)) ~ (~{~,))*.

Step 3. Fix p < n, and write n! =ptm, where p.~m. For each g ~ S,,

m ~ , # ( d ) d

~o(m) " Z :d[m, (d, [gl) = 1 �9 g + (elts of smaller order).

]he matrix for ~,,p is thus triangular, and the diagonal entry for g is:

qg(m-~'I-[{q-ql--:P~=q<n,q/Hg]} = 1 - I {~ :P+q l l g [ }

iwhere q is always prime). So if we define

aq(n) = # {conjugacy classes of g ~ S, with q[ [g[}

for primes q_< n, then / q \~.(")

det(7~. ~)= ~ [ a ~ ] . (4) ' q = n \ q - - , /

q4:p

60 R. Oliver

Step 4. Now set

k = r k R e ( S . ) = # {conjugacy classes in S.}.

By [20, Theorem 10], for each p,

K](~,vS.) [ } 1 ~-Kl(~pSn/J) ,

where J is the Jacobson radical. Write

~pS. /J = ~-I M,,(F(P~')) /=1

for some m, ri, sv Then m is the number of distinct simple lFp[S,]-modules, and so by [2, Theorem 21.25]:

m = # {~Zp-conjugacy classes of g e S. with P,flgl} = k-o~p(n).

Similarly, if F____Fp is a splitting field for ~pS./J, then

s~ = ~ {conjugacy classes of g e S. with p4V[gl} = k - %(n) = m. i=1

Thus s~ = 1 for all i, and

K'~ (~,S . ) [1 ] ~_ K l (~,pSn/ J) ~ ( ~ / p - l ) k - ~p(n). (5) LP3

Upon combining (3), (4), and (5), we now get, for each p,

IK'~(~,)/K'~(:~pS.)[ = I A ~ ( S . ) r �9 1-[ ( q - 1)~; q(")" (p - 1) ~p(") �9 qNn

After taking the product over all p, the last two terms cancel, leaving

l--[ [K,(~p)/K](:~,S.)I = [A~(S.)I. [] p <=n

For convenience, for any k > 1 and any prime p, we write ordv(k) = m ifp" is the largest p-power dividing k. For n > 1, p(n) will denote the number of partitions of n; i.e., the number of conjugacy classes in S..

Proposition 2.3. Fix n > 2, and let g l . . . . . gp~,) be conjugacy class representatives for elements of S.. For each prime p, set

ap(n) = 41= { 1 < i < p(n) : p[ [gi[} ;

Then

tip(n) = ordp [,=lfl,

ordpID(7ZS.)]=�89 ~, ordp (q - 1).~q(n) if p > 2 p<q.<n q prime

=2-p(n)+�89 ~ ordz(q-l).O~q(n) if p : 2 . 2<q<n q pr i m e

Logarithmic Descriptions of K'I(~vG) 61

Proof. Let 9Jl ~ES, be a maximal order; n ! 9J/_cZS, by [14, Theorem 41.1]. By [7, Theorem 3.3], the pullback squares

Z S . - ~ 9J~

1 l induce an exact sequence

p<-n [ p<n 1 f

ZS./(n!~) , ~ / ( n ! ~ )

0 KI(TZS,)~KI( f f2d)--*Coker(K,( f ) )~D(ZSv)- ,O

and an isomorphism Coker (Kl ( f ) ) ~ M (KI(~ilv)/KI(~vS.)) (note that Ko(~pS,) p<n is torsion free by [15, Theorem 6.1]). Hence

[D(ZS.)[ = [prI=<. IKl(~llp)/K'~(~pS.)l] ~ IK'~(gS.)I/IKIOJI){. (1)

Recall that 93l is a product of p(n) matrix algebras over 7Z,. It follows that IK 1(93l)1 = 2 p("), and that K'I(TZS.) is finite. By a theorem of Wall [21, Theorem 6.1]:

to rsK;(TZS.) -{+ l} x (S.)"b~Z/2 x2~/2 (n=>2).

So by (1) and Proposition 2.2,

ID(ZS.)I = 2 2- v~")IA~(S.)[. (2)

Since QS. is a product of matrix algebras over Q, the formula in [10, Theorem 11] takes the form

t_' =~ [ ~ ] ) ~ o ( ] g , ] ) , ` ~ " ~1/2 IZ~(S.)l = lw tg,)/g,] ] �9

For any g e S,, JN(g)/Z(g)[ = ~o([g[) (all generators o f ( g ) are conjugate to g); and so this reduces to

IA,~(S,,)I-- L~: l ~ - _ 1 ki = 1

Lps,K P / A [_p<__n J

Fhe formulas for ordv]D(ZS,)] follow immediately from (2) and (3). []

It remains only to describe the %(n), fir(n) defined above. As usual, we set p(0) = 1.

Lemma 2.4. For any r > O, set

3k 2 + k 3k 2 - k ( - 1 ) k-I if r = - - o r ( a n y k e Z , k>O)

d ( r ) = / 2 0 otherwise.

62 R. Olivcr

For k > 0, let ~o(k) denote the number of divisors. For n > 0 and primes p < n, let 0~p(n), tip(n) be the numbers defined in Proposition 2.3. Then

nip

~p(n)-- 32 f ( k ) p ( n - p k ) and tip(n)= ~ ordp(k)cro(k)p(n-k ). k = l k = l

Proof. Step 1. To describe ap(n), it is convenient to work in the power series ring 7Z[[x]]. Set

P(x) = p(n)x"= I-I n=0 k = l \ l - - X f

For any p and n, ~p(n) is the number ofconjugacy classes in S, of elements of order a multiple of p; i.e., the number of partitions of n containing a summand divisible by p. So

n = l k = l k ? l P ( x k)

= P ( x ) [ 1 - - k ~ (1--xPk) ]"

By Euler's pentagonal identity (see, e.g., [6, Sect. 256]):

k = l k = l

Hence Y'. ~p(n)x" = P(x) (Z f (n) xp"); and for each n: n/p

ap(n)= 32 f ( k ) p ( n - p k ) . k = l

Step 2. If g e S,, as a product of disjoint cycles, contains sk k-cycles for 1 _< k_< n, then

IZ~.(g)l= 1-"1 (k sk' (sk)O-- 1;I ~l (rk). k = l k = l r = l

For each r, k > 1, these are exactly p(n - rk) partitions of n containing at least r k's. Hence, if gl . . . . , gpt,) are conjugacy class representatives for elements of S,, then

p(n) nlk

r I IZsn(g,)l = l~I I-I (rk) p`"-'k)= [I k ~~ i=1 k : l r : l k = l

So for any prime p,

[- P(") 1 fl.(n)-=ordpL,_H ' IZs.(e,)i = k=, ~ ~176 []

Proposition 2.3 and Lemma 2.4 now combine to give:

Theorem 2.5. For rational r > O, set

l 3k 2 - k 3k 2 + k ( - 1 ) k-1 if r = - - o r (k~72, k>O)

f (r) = 2 ~ - - 0 otherwise (including all r qD 22).

Logarithmic Descriptions of K](~pG) 63

For any prime p and any k > 1, define

ap(k)=�89 ~, (ordp(q-1) . f ( k /q ) ) . p<q<k qprime

Then for any prime p and any n >___ 2:

I k~:l ap(k)p( - ) if p is odd ordplO(ZSn) l

= [ 2 - p ( n ) + k~l az(k)p(n-k) if p = 2 .

Proof. Immediate. []

This formula is much easier to apply than it first looks: f (r ) is mostly zero for large r, and so most terms in the formula for ap(k) vanish. For small values of n, Theorem 2.5 gives the following values for ID(7ZS,)I"

n 5 6 7 8 9 10 ]D(ZS,)I 2 24 26.32 216.33 224-38 246.311.5

Lemma 2.1 can of course be used to obtain more precise information about D(7/S,). For example, the maps 6,,p, for primes p < n, can be shown to induce an epimorphism

2A~(S.)--~ [ I [Kl(9~p)/(tors K l (~J~p) q- K ' 1 (~pSn) ) ] . p<n

This can then be used to compare exp(D(~S,)) with the Artin exponent a~(S,) -exp(A~(S,)). By results of Lain [4],

a~(S,)=(n!) / I-I P iv<=,

whenever n > 5. The above epimorphism, together with results about the Swan subgroup T(ZS,)C=D(ZS,) [16, 18] then show that for any n,

exp(D(ZS.))=�89 or a~(S,)

{and it seems likely that the first case holds when n >4). For example, if p is odd, then together with Theorem 2.5 this implies that

D(ZS,)tp ) ~- 0 if n < 2p

~- 7Z/p if n = 2p

~_ (7l/p)r~. 2p) if 2p < n < 3p, and 2p + 1 not prime

~_(~/p)pt, 2p)+ pc, 2p- ~) if 2p < n < 3p, and 2p + 1 is prime.

References

I. Curtis, C., Reiner, I.: Representation theory of finite groups and associative algebras. New York: lnterscience 1962

2. Curtis, C., Reiner, I.: Methods of representation theory, with applications to finite groups and orders, Vol. 1. New York: Wiley 1981

64 R. Oliver

3. Dress, A.: Induction and structure theorems for orthogonal representations of finite groups. Ann. Math. 102, 291-325 (1975)

4. Lam, T.-Y.: Artin exponent of finite groups. J. Algebra 9, 94-119 (1968) 5. Lang, S.: Algebraic number theory. New York, Basel: Marcel Dekker 1977 6. Macmahon, P.: Combinatory analysis. Chelsea 1960 7. Milnor, J.: Introduction to algebraic K-theory. Princeton: University Press 1971 8. 0liver, R.: SK 1 for finite group rings II. Math. Scand. 47, 195-231 (1980) 9. Oliver, R.: SK1 for finite group rings III. Lect. Notes Math. 854, pp. 299-337. Berlin,

Heidelberg, New York: Springer 1981 10. Oliver, R.: D(ZG) + and the Artin cokernel. Comment. Math. Helv. 58, 291-311 (1983) 11. Oliver, R.: Lower bounds for K~P(~pn) and K2(7/n ). J. Algebra (to appear) 12. Oliver, R.: An exact sequence involving Kl(2pn ) and K2(~.pn). Lect. notes Math. 1046,

pp. 255-260. Berlin, Heidelberg, New York: Springer 1984 13. Oliver, R.: The Whitehead transfer homomorphism for oriented S~-bundles. Math. Scand. (to

appear) 14. Reiner, I.: Maximal orders. London, New York: Academic Press 1975 15. Swan, R.: Induced representations and projective modules. Ann. Math. 71,552-578 (1960) 16. Taylor, M.: Locally free classgroups of groups of prime power order. J. Algebra 50, 463487

(1978) 17. Taylor, M.: The locally free classgroup of the symmetric group. Ill. J. Math. 23,687-702 (1979) 18. Ullom, S.: Nontrivial lower bounds for class groups of integral group rings. Ill. J. Math. 20,

361-371 (1976) 19. van der Waerden, B.: Algebra. Berlin, Heidelberg, New York: Springer 1966 20. Wall, C.T.C.: On the classification ofhermitian forms, III: Complete semilocal rings. Invent.

math. 19, 59-71 (1973) 21. Wall, C.T.C.: Norms of units in group rings, Proc. Lond. Math. Soc. 29, 593 632 (1974)

Received May 6, 1985