Communicated by M. F. NewmanFor formations of finite soluble groups, the properties of Frattini closure and local defineability are known to be equivalent (see [2] It is clear from the example of Barnes and Gastineau-Hills mentioned above, that the analogous proposition to Theorem 1 is false for finite-dimensional soluble Lie algebras.The terms "formation", "local" and "Frattini closed" used above have natural definitions analogous to those used in group theory. Thus, following usual terminology, we refer to a class of rings as a homomorph whenever it contains all homomorphic images of its members and as a formation if in addition it is subdirect product closed. For any ring R, the intersection (f)(R) of its maximal ideals, when such exist, is called the Frattini subring of R. It is well-known and * A prime is said to divide a formation of finite rings if it divides the characteristic of at least one non-trivial ring in that formation.
375use, available at https://www.cambridge.org/core/terms. https://doi
To Bernhard Hermann Neumann on his 60th birthday
Communicated by G. E. WallA group is called an s2I-group if and only if it is locally finite and all its Sylow subgroups are abelian. Kovacs [1] has shown that for any integer e the class s3t e of all s3I-groups of exponents dividing e is a variety. Little is known about the laws of these varieties; in particular it is unknown whether they have finite bases. Whenever s2l e is soluble it is an easy matter to establish explicitly a finite basis for its laws namely the exponent law, the appropriate solubility length law and all laws of the type [x m , y m ] m where e = p a m, p is a prime and p does not divide m. (The significance of the last type of law is made clear by Proposition 2 below and the obvious fact that any group that satisfies a law of this type for given prime p has abeh; n Sylow ^-subgroups.) For e less than thirty s3l e is clearly soluble whilst PSL(2, 5), the non-abelian simple group of order 60, is contained in s9I 30 so that the case e = 30 is, in a sense, the first non-trivial case to be consider, d.The purpose of this note is to establish the following set of laws as a basis for the variety s2t 3O :(i) z 30 (ii) {((a; 6 */ 12 ) 5^6 */ 18 ) 5 ) 3^6 . y 6]e}6 (iii) ((a; 1( y 0 ) 6 [> 10 >2/ 10 ] 2 ) 10 (iv) [ O e 0 X 0 ] , [ M 6 0 X 0 ] Z ]
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