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R. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic(R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H.

Let A be a quasi-Frobenius algebra over a field K. A has a complete (co)homology theory which may be established upon an augmented acyclic projective complex, i.e. a commutative diagram of A-double-modules with exact horizontal row, projective Xp, and with epimorphic resp. monomorphic ε and t. Negative-dimensional cohomology groups, over an A-double-module, are expected to be in close relationship with (ordinary positive-dimensional) homology groups.

A short proof is given for Mine's partial converse to a fundamental theorem of Frobenius on irreducible nonnegative matrices.

(1969). A Generalization of a Result of Frobenius. The American Mathematical Monthly: Vol. 76, No. 8, pp. 924-925.

Consider the following result.PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.

(1970). Open Mappings and the Fundamental Theorem of Algebra. Mathematics Magazine: Vol. 43, No. 1, pp. 39-40.

For a commutative k-algebra B, consider the subalgebra (B?n)Sn of the nth tensor power of B, formed by the tensors invariant under arbitrary permutations of the indices. Necessary and sufficient conditions are found for (B?n)Sn to be Frobenius. When dimk B $ 2, these say that B is Frobenius and n! is invertible in k, unless B is separable. Some additional cases occur for two-dimensional algebras in positive characteristic, depending on the divisibility of n + 1.

(1980). A Proof of the Frobenius Rank Equality. The American Mathematical Monthly: Vol. 87, No. 6, pp. 481-482.

A conjecture of Frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups. Let G be a finite group and n be a positive integer dividing IGI. Let Ln(G) = {x E G xn = 1). Then by a theorem of Frobenius [6] one knows that ILn(G)I = cnn for some integer cn. Frobenius conjectured that Ln(G) forms a subgroup of G provided ILn(G)I = n (see [2]). Zemlin [25] has reduced the conjecture to the classification of finite simple groups which is now complete (see [8]). The author has verified the conjecture for the Fischer Griess monster F1 and the Fischer baby monster F2 in [24]. The purpose of this note is to prove the following THEOREM. The conjecture of Frobenius is true for all the sporadic simple groups. The proof of our theorem has been carried out in the following way with the use of a computer. Let G be one of the sporadic simple groups. By [24] we may assume that G # F1 and G # F2. Let f(G, t) be the number of elements of order t in G and Ord(G) = {order of x I x E G}. Tables of f(G, t) are given in the Appendix; see the supplements section at the end of this issue. For f(G, t) the reader is referred to the following papers: Ml1, M22, M23 Burgoyne and Fong [1] M12, M24 Frobenius [5] ii Janko [14] HJ = J2 Hall and Wales [8] HJM= J3 Janko [15] J4 Janko [16] HiS Frame [4] Suz Wright [23] McL, .3 Finkelstein [3] Rud Rudvalis [19] HHM Held [10] LyS Lyons [17] Received June 21, 1982; revised March 19, 1984 and July 11, 1984. 1980 Mathematics Subject Classification. Primary 20D05.

then jT"_1 is a Hankel matrix for all Toeplitz matrices TnI' Thi s proced ure, howe ver, does not carry Hermitian ToepLitz matrices to Hermitian (i.e., real) Hankel matrices. The theore m of FischerFroebenius asserts that a class of tran sformation s exist each of which uniformly carries Toeplitz matrices to Hankel matrices in such a way that Hermitian Toeplitz matrices are carried to Hermitian Hankel matrices. Recently I. C. Iohvidov has publi shed three proofs of this result. One of these proofs is a direct but somewhat intricate calculation; it may be found on pages 211-213 of [1]1. A second proof, to be found on page 217 of [1] and also in [2], makes a preliminary reduction to the case of positive definite ToepLitz matrices, then takes advantage of a decomposition of definite Toeplitz matrices known from the theory of the trigonometric moment problem. The third proof, in [2], avoids the .reduction to the positive definite case, and uses instead a more co mplicated decomposition of Toeplitz matrices due to Iohvidov and Krein [3, p. 338]. The purpose of this paper is to give a short and direct proof of the Fischer-Frobe nius theorem. Our proof is based on a simple decomposition of arbitrary Toeplitz matrices, for which the proof is almost a triviality and whic h was apparently not noticed in [1] and [2]. See equation (3). Iohvidov's techniques then may be applied to (3) to produce the des ired result rapidly.

In the development of local class field theory by means of the theory of algebras, the important norm transfer theorem is essentially equivalent to the following theorem on cyclic algebras: (C1L/L, l'l, Al) (C2L/L, 2 A2) implies (C,/k, ol, NL/kAl) (02/k, 02 NL/kA2) whenever Ci/k are cyclic, L/k separable algebraic, o, and au generating automorphisms for Cijk and CL/L respectively, and Ai e L. Chevalley [2], Nakayama [6], Hochschild [4], and Schilling [7, Theorem 7, p. 166] do the equivalent of proving this, with k restricted to be a p-adic field and the "implies" replaced by "if and only if". All these proofs depend on the proposition that for p-adic k every algebra class over L is obtained by extending the center of an algebra class over k. This is by no means true for general fields k: for example, if k is a number field instead of a p-adic one, the theory of invariants of algebras gives counter examples at once for this proposition, as well as for our theorem with "implies" replaced by "if and only if". So it is of interest to prove the theorem without any restriction on the field k; Section 2 contains this proof. Section 1 contains a new notation which simplifies Section 2, and also has the advantages of removing the awkwardness of the usual definition of cyclic algebras and of clarifying the relation between algebras and group characters.

In the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by Frobenius's notion of induced representation. To every representation L of the subgroup G of the finite group 65 there is assigned a well defined representation UL of ( called the representation of M induced by L. In I of this series [10] the author began a systematic study of a generalization of this notion in which (5 is a separable locally compact topological group and the spaces of L and UL are possibly infinite dimensional Hilbert spaces. In particular [10] contains a generalization of the Frobenius reciprocity theorem; that is the theorem asserting that UL contains the irreducible representation M of (D just as many times as the restriction of M to G contains the irreducible representation L of G. The generalization contained in [10] is unsatisfactory in that it deals only with the discrete finite dimensional irreducible components of the representations concerned and becomes vacuous when these representations decompose in a continuous fashion or have no finite dimensional irreducible components. A more satisfactory generalization has been obtained by Mautner [13, 14]. It deals in a consequent fashion with continuously decomposable representations of 5 but is restricted by the requirement that G be compact. This means in particular that only discretely decomposable representations of G need be dealt with. The principal result of the present article is a generalization of the Frobenius reciprocity theorem which deals effectively with continuously decomposable representations of both M and G. Since in compensation for the hypothesis that G be compact, we require that both G and 5 have regular representations which are of type I, our theorem does not quite include that of Mautner. However we show in addition that whenever the regular representation of G is not only of type I but also discretely decomposable then the requirement that the regular representation of (M be of type I can be eliminated. Thus our methods also yield a result which includes that of Mautner. In neither of our results is it necessary to assume that the groups concerned are unimodular. A noteworthy feature of our principal theorem is that it includes a reciprocity not only for the multiplicities but for the measures involved in the continuous decompositions as well. The basic idea in our approach is a new proof of the Frobenius reciprocity theorem in the finite case which has the advantage of generalizing significantly

The celebrated orthogonality relation for the coefficients of the regular representation of a group was extended first to the modular case by Nesbitt, and then to Frobenius-algebras by the writer; the proof was reproduced in [4]1 together with a second proof. Another proof of this orthogonality, for the coefficients of the regular representations of Frobenius-algebras, and its interesting application to faithful representations were given by Brauer [1]. In the present note we propose a still different proof, and generalize the orthogonality to quasiFrobenius-algebras. 1. A class of automorphisms in a Frobenius-algebra. Let 21 be a Frobenius-algebra [1; 2; 3; 4.] over a field Q, and let (1) (a1, a2, n, a) with aa ar =E as be its basis. Let