Can. ]. Math., Vol. XXIII, No.3, 1971, pp. 541-543

ON CLASS SUMS IN p-ADIC GROUP RINGS

SUDARSHAN K. SEHGAL

;.

1. Introduction. In this note we prove that an isomorphism of p-adicgroup rings of finite p-groups maps class sums onto class sums. For integralgroup rings this is a well known theorem of Glauberman (see [3; 7]). As anappliCation, we show that any automorphism of the p-adic group ring of afinite p-group of nil potency class 2 is composed of a group automorphismand a conjugation by a suitable element of the p-adic group algebra. Thiswas proved for integral group rings of finite nilpotent groups of class 2 in [5].In general this question remains open. We also indicate an extension of atheorem of Passman and Whitcomb. The following notation is used.

G denotes a finite p-group.Z denotes the ring of (rational) integers.2p denotes the ring of p-adic integers.Qp denotes the p-adic number field.K denotes Qp, the algebraic closure of Qp which contains A the field

of all algebraic numbers.denotes the group ring of G with coefficients from Zp,denotes the class sums of G.denotes the class sums of H.

denotes the primitive central idempotents of Qp(G).denotes the number of elements in ith conjugacy class of G and Hrespectively.

{Xi} denotes the absolutely irreducible characters of Qp(G).denotes the degree of Xi'

Zp (G){Ci}{KtJ{ei}hi, ki

.~

Zi

2. Theorem of Glauberman. We state our main theorem.

THEOREM 1. Let e: Zp(G) ~ Zp(H) be an isomorphism. Then e(Ci) = ::l:Ki

for all i.

Proof. Replacing e(G) by G we can assume that Zp(G) = Zp(H). We haveto prove that Ci = :!=Ki for all i. At first we claim that

(1) Ki = L aijC}j

with ai; E Z.

We know (see [1; p. 236]) that

(' (2) e - Zi '"i - (G : 1) L.t Xi(go)Co, go E Co,

Received November 3, 1970. This work was supported by N.R.C. Grant No. A-5300.

~ 541

542 SUDARSHAN K. SEHGAL

and that

(3) Ci = L: hixv(gi)Zeo,

v vgt E Ci.

By the same token we have

(4) K. = '" k Xi(Xj) e1 .t: J jZ

t,t t

Xj E Kj.

Substituting the value of et from (2) in (4) we obtain

(5)1

Kj = (G : 1) fj kjXi(Xj) Xi(gv) Co'Also,

(6) Kj = L: ajlC, for some aj' E Zp.,Comparing (4) and (5) we have

1 '" -(7) ajv = (G : 1).Lt kjxt(xj) Xt(gv)'

It follows from (7) that (G: 1) ajv is an algebraic integer. Since the pmthcyclotomic polynomial over Qp is irreducible (see [2, p. 212]), by taking traceQp(~)/Qp where ~is an appropriate root of unity, we get from (7) that (G: 1) ajflis a rational number and hence a rational integer. But since ajv is a p-adicinteger and (G : 1) is a p-power it follows that ajv is a rational integer. Hence(1) is established. Now we use the argument of Glauberman to conclude thatajfl = ::I::Ojv'This argument consists mainly of assigning a weight

~.

W(Kl, . . . ,Km) = L: Xt(Kj) Xt(Kj)i.J

to class sums of every group basis H and observing that

W(Kl, . . . ,Km) = (G : 1) L: kpi/ > (G : 1)2,i.j

with equality if and only if for each i there is exactly one j such that aij ~ 0and for that j, aij = ::I::1. Hence the class sums of any group basis H haveweight (G: 1)2 if and only if they are precisely {::I::Ci}. Reversing the roleof G and H one obtains that the only class sums of a group basis with weight(G: 1)2 are precisely {::I::Kd. It follows therefore that {::I::Ci}= {::I::Ki}.

3. Applications. We state two applications and indicate the proofs brieflyas they are well known in the integral case and the proofs in this case areidentical.

THEOREM2. Let 0 be an automorphism of Zp(G), where G is nilpotent of class 2.Then thereexists an automorphismXof G and a unit l' of Qp(G) such that t

0 (g) = ::I::'Yg)"}'-lfor all g E G.

!.

CLASS SUMS 543

Proof. As in [5], the Theorem follows from Propositions 1 and 2.

PROPOSITION1. Let 0 be an automorphism of I(G) where I is an integraldomain with field of quotients F. Suppose that O(Ci) = C/, and that thereexistsan automorPhism q of G such that q(Ci) = C/, for all i. Then we can find aunit'Y E F(G) such that

O(g) = 'Yg""Y-Ifor all g E G.

PROPOSITION2. Let P.be an automorPhism of Zp (G) where G is nilpotent ofclass 2. Suppose that p.(Ci) = C/,for all i. Then there exists an automorPhismq of G which, when extended to Zp(G), satisfies q(Ci) = C/, for all i.

Proof. Proposition 1 has been proved for Z (G) in [5] but the proof is thesame for any I(G). For Proposition 2, the existence of such a q is provedin [6]. That q(Ci) = C/, follows just as in [5].

Passman and Whitcomb [3; 7] proved the next Theorem for Z(G).

THEOREM 3. Let 0: Zp (G) ~ Zp (H) be an isomorphism. Then there exists a1 - 1 correspondence N ~ cf>(N) between normal subgroups of G and H. Thiscorrespondence satisfies

(1) NI C N2 {:=} cf>(N1)C cf>(N2)(2) (N: 1) = (cf>(N): 1)(3) (NI, N2) = (cf>(N1), cf>(N2».

Proof. The correspondence is established due to Theorem 1. The proofs of(1) and (2) are trivial, and (3) follows as in [4].

REFERENCES

1. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras(lnterscience, New York, 1962).

2. H. Hasse, Zahlentheorie (Academie-Verlag, Berlin, 1963).3. D. S. Passman, IsomorPhic groups and group rings, Pacific J. Math. 15 (1965),561-583.4. R. SandHng, Note on the integralgroup ring problem (to appear).5. S. K. Sehgal, On the isomorphism of integralgroup rings. I, Can. J. Math. 21 (1969), 41Q-413.6. S. K. Sehgal, On theisomorphism of p-adic group rings, J. Number Theory 2 (1970), 500-508.7. A. Whitcomb, The group ring problem, Ph.D. Thesis, University of Chicago, Illinois, 1968.

University of Alberta,Edmonton, Alberta

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