Admissible Groups, Symmetric Factor Sets, and Simple Algebras

Let K be a field of characteristic zero and suppose that D is a K-division algebra; i.e. a finite dimensional division algebra over K with center K. In Mollin [1] we proved that if K contains no non-trivial odd order roots of unity, then every finite odd order subgroup of D* the multiplicative group of D, is cyclic. The first main result of this paper is to generalize (and simplify the proof of) the above. Next we generalize and investigate the concept of admissible groups. Finally we provide necessary and sufficient conditions for a simple algebra, with an abelian maximal subfield, to be isomorphic to a tensor product of cyclic algebras. The latter is achieved via symmetric factor sets.

component of the group algebra KG for some finite group G.We let [A] denote the equivalence class of the K-central simple algebra A in B(K).The notation A B means [A] [B] in B(K).When A (R) B is written, the tensor product is assumed to be taken over the algebra in the left factor.For most basic results pertaining to S(K) the reader is referred to Yamada [2].
A crossed product algebra will be denoted (K/k,B) which is the central simple k-algebra having K-basis uo with o e G(K/k), subject to: uouT (o,T)uo and ux x u where x e K and o,T G(K/k), the Galois group of K over k.For further information pertaining to crossed products the reader is referred to Relner [3].
Finally we comment on notation.If m is a positive integer with m pa n where the prime p does not divide n then Iml p i.e.Iml denotes the p-part of m.A th P P primitive m root of unity will be denoted by m 2. SUBGROUPS OF SIMPLE ALGEBRAS.
Let K be a field of characteristic zero.The major thrust of this section is to provide a generalization of Mollin [i, Theorem 3.6, p. 243].To pave the road we first need a definition and some preliminary results.
Let D be an E-division algebra for some field E, finite dimensional over K, and let n be a fixed positive integer.We say that D is (n,K)-adequate if there ests a K-division algebra B with K I c D c M (B) where I is the identity of Mn(B), te full n ring of n n matrices with entries from B.
We need the following result which generalizes Mollin [I, Theorem 3.4, p. 242].In what follows Aut(D) denotes the automorphsm group of D. THEOREM 2.1.Let E/K be finite Galois.If D is an (n,K)-adequate E-dlvislo algebra and e G(E/K) then extends to Aut(D).
PROOF.Let A M (B) where D is embedded in A. If C CA(E) denotes the cerra- n lizer of E is A then by Relner [3, Corollary 7.14, p. 96] we have C] The remainder of the proof follows from this juncture exactly as the proof of Mollln .,The following which is immediate generalizes Mollin [i, Corollary 3.5,p.242].COROLLARY 2.2.Let E/K be finite Galois, and let D be an (n,K)-adequate E-division algebra If [D] e S(E), with exponent m, then e is in K.
In what follows a subgroup of Mn(D) will mean a finite multpl=atlve subgroup of M (D).The following result is immediate from Hikari [4, proposXZns 1 and n pp.369-370] Note that if G is an abelian group expressible as a [rect sum of cyclic groups C ... C with ICil e i such that eilei+ for i I, m-I and n e # 1 then we say that G has invariants of length n. n LEMMA 2.3.Suppose that G is a subgroup of Mn(D) where n p fz he n[nimum odd prime divisor p of IGI.Then all odd Sylow subgroups of G are aelian with invariants of length less than or equal to n.Now we need another definition.Given a K-division algebra D a a fixed positive integer n, suppose that G is a subgroup of Mn(D).We let () {Z aigi:a i e Q'gi' e G}.V(G) is a Q subalgebra of Mn(D) and is, i fact, a direct summand of the group algebra QG.This generalizes the concept as used In Andtsur [5] for the case n 1 wherein V(G) is a minimal division algebra eonaunng G. Now we are in a position to prove the main result of this section.
THEOREM 2.4.Let n be a fixed positive integer and let De a K-divlsion algebra.
If K contains no non-trivial odd order roots of unity then every odd order subgroup of M (D) with n < p for the minimum prime divisor of IGI is abelan with invarlants of length n.
PROOF.Suppose V(G) M (Di) c Mn(D).Thus Z n < < p.But n i dvlde& i n.
i i--IGI, (see Curtis and Reiner [6, hapter IV]).Hence, n i l for each is commutative for each i then G is abelian in which case we get the result from lemma 2.3.Thus we assume that D. is a (non-trivial) division algebra for some 3 Then [Dj] has odd exponent, r > 1 say, in S(E) where E/Q is finite abelan Amitsur [5]).But D. c M (D) so D. KE is (n,K)-adequate, and D. (R) KE is a division 3--n 3 algebra such that [Dj @ KE] has exponent r in S(KE).By corollary 2.2, er is in K, contradicting the hypothesis. Q.E.D.

ADMISSIBILITY
The following definitions generalize concepts introduced in Schacher [7].Let K/k be a finite extension of fields.K is called (n,k)-adg.quateif and only f K is a self-centralizing maximal subfield of M (D) for some k-division ring D, and n is the smallest positive integer for which there is such an embedding as k-algebras, (see Reiner [3, Chapter 7]).A finite group G is called (n,k)-admissible if and only if there is a Galois extension K of k with G G(K/k) and K is (n,K)-adequate.
G is called totally n-admissible if and only if for each par of number field K and k with K Galois over k and G G(K/k) we have that K Is (n,k)-adequate.
The reader should note hat the concept of K-adequacy was used in [i] to prove that every finite odd order multiplicative subgroup of a division ring D is cyclic, whenever the center of D has no non-trivial odd order roots of unity, ([I, Theorem 3.6, p. 243]).Herein our extension from the K-adequacy concept to the (n,K)adequacy concept tacitly allowed us to prove theorem 2.4which generalized [ibld] since we were in a position to consider subgroups of M(D) for a gven fixed positive integer n.The mechanism for proving theorem 2.4 was the Q-subalgebra V(G) of M (D).This mechanism opened the door for the use of lemma 2. (see Janusz [9, Chapter IV]).Define [A (p)] e B(k) by Invpi((P) ) (-l)I/lol for i 1,2; and inv (A (p)) 0 for all q # Pi" q Now for each pllGl form such a A(P) and let be the k-division algebra (P).Then K splits and by Reiner [3, 28, pp.237-241] n is the smallest p G positive integer such that K is a self-centralizing maximal subfield of M (4); i.e.K is (n,k)-adequate.Hence G is totally n-admissible.
Conversely suppose that G is totally n-admlssible.Then if G G(K/k) we have that K is embedded in M (D) for some k-division ring D, wth n being the smallest n such positive integer.By Reiner, [3, ibid] we have that n IK:kl:I.Moreover, by Artin and Tare [I0, p. 75] we may assume that K/k is unramifled.Now by Albert, [II, thoerem 33, p. 150] we have is equal to the least common multiple of the degrees l:kel taken over all primes at which D has non-zero for each Masse invariant.Thus there exists a K-prime such that IK:kpIp__ ID:klp p dividing IK:kl.Since G(/) is generated by the Froebenius automorphism of for in K/k then G must have an element of order IK:klp/Inlp IGIp/Inlp each p dividing IGI. Q.E.D.
The following are immediate.COROLLARY 3.2.Let n divide IGI.If G is totally n-admissible then G is metacyclic.In particular G is solvable.R.A. MOLLIN COROLLARY 3.3.For each pair of number fields K and k with K/k normal and G G(K/k) we have that the following are equivalent: (i) IGl , the l.c.m, of the inertial degrees of all k-primes in K/k.
(2) All Sylow p-subgroups of G are cyclic.
(3) G is totally -admlssible nilpotent (4) Let m,r be two relatively prime integers.Pu-t s (r-l,m), t m/s and n no Now we consider the case where G is abellan.
THEOREM 3.4.Let G be a finite abelian group.Then G is (n,k)-admlsslble for some number field k and some positive integer n dividing IGI.
PROOF.Schacher [7, Theorem 6.2, p. 465] guarantees that G G(K/k) for some abelian extension K of k.We must show that K is (n,k)-adequate.Since C is abellan then there exists a subgroup of order n.Let E be the fixed field of this subgroup.
E is embedded as a maximal self-centralzing subfield of a k-dlvlsion algebra A.
Since K splits A then by Relner [3, Chapter 7] K is embedded in M () as a self- n centralizing maximal subfield with n as the least such positive integer.
Q.E.D. Now suppose that G is finite abelian and M () Is a K-central simple algebra n with G G(L/K) where L is a maximal subfield of M ().Thus M (A) (L/K,8) for a n n sultable factor set 8. Now G <Ol>X<o2>x...x<o t> where o i has order nl, say.
It is natural to ask whether (L/K,8) has a similar decomposition as a product of cyclic algebras of exponent n i.
The following theorem yields a necessary and sufficient condition for such a decomposition to exist.In what follows a symmetric factor set 8 means one such that 8(o,z) 8(z,o) for all o,z e G. Moreover L i shall denote the fixed field ji <oj>" THEOREM 3.5.Let G be a finite abellan group.Then G is (n,K)-admlsslble for some number field K and some n dividing IG[.Suppose that A (L/K,8) is a K-central simple algebra with G G(L/K).Then A A ...@ A t where each A i is cyclic K-central simple with maximal subfield L i if 8 is symmetric.Conversely we have the weaker result: If A A (R)...(R) A t then is cohomologous to a symmetric factor set.
PROOF.Assume 8 is symmetric.Now each E G clearly extends to an inner -i au ;a e A. Since (,) (,) for each , E C then automorphism of A:a / uo u u for each , E G.These two facts imply that hiJ h i for each i and j uou T o where h i oi.Therefore h i c K for each i, which implies A i (Li/K,hi) is a cyclic crossed product algebra with maximal subfield L i. Xl xl then Now since L L x L 2 (R)... L t then n n n2...n t.Therefore the map from A to A (R)... (R) A t given by u / u xl (R)..  h is symmetric, so inf(i)h is symmetric.Since the factor set of inf(1)Al inf(t)At is the multiplication of inf(i)hi it follows that 8 is cohomologous to a symmetric factor set.
Q.E.D. ACKNOWLEDGEMENT.The author's research is supported by N.S.E.R.C. Canada.
ar> where IGI mno and g.c.d. (n o,t) I.
Conversely assume A A (R)...A t Now byReiner [3, Theorem (29.16),p.249] A (L/K, inf(i)h where inf (i) denotes the inflation map from L to L Moreover i i i