This paper originates from the following observation: The standard construction of the symmetric (or antisymmetric) algebra of a module E over a commutative ring K with unit is carried out in two distinct stages. One first defines the pth symmetric (or antisymmetric) power for each p E N by using the natural representation of the symmetric group S, on the pth tensor power Op E, and then factoring out a submodule, defined by the unit character (or the signature). Then one checks that the direct sum of the modules thus obtained inherits an associative product from the tensor algebra over K: QE= UpDO BP E. Clearly, the first step can be mimicked starting with any sequence of permutation groups W = ( W, w What are, then, the condition on W and x, which would guarantee that the direct sum [x] E= IJ,,0 [xl” E inherits a structure of associative algebra from @E? The answer turns out to be very explicit, and involves an interesting endomorphism, w, of the symmetric group S, of permatuations of N with finite support. The precise statement is given in Theorem 1.3.1, which, in particular, shows that for [x] E to be an associative algebra, the sequence W has to satisfy for each p B 1 the conditions W, < W,, 1 (with respect to the natural embedding of S, in S, + ,), 4Wp)G wp+JY while the sequence x has to satisfy xP+ ,, wP = xP and xP + i 0 w, wP = xP. Since the converse is also true (that is, such a sequence of characters does define an associative algebra [x]E), the next question which comes up is to classify all such sequences. This is done in Theorem 1.3.3; it shows that the group of these sequences (with componentwise multiplication) is naturally isomorphic to the trivial group, or, to the subgroup Z(K) of the group U(K) of units of K, consisting of all involutions. In fact, this isomorphism is constructively defined, starting 479 0021~8693192 $5.00