A basis for the laws of the variety ѕU30
1969; Cambridge University Press; Volume: 10; Issue: 3-4 Linguagem: Inglês
10.1017/s1446788700007771
ISSN1446-8107
Autores Tópico(s)Crystal structures of chemical compounds
ResumoA group is called an ѕ U -group if and only if it is locally finite and all its Sylow subgroups are abelian. Kovács [1] has shown that for any integer e the class ѕ U e of all ѕ U -groups of exponents dividing е is a variety. Little is known about the laws of these varieties; in particular it is unknown whether they have finite bases. Whenever ѕ U 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 α m , p is a prime and p does not divide m . (The significance of thelast 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 abelirn Sylow p-subgroups.) For e less than thirty ѕ U e is clearly soluble whikt PSL(2, 5), the non-abelian simple group of order 60, is contained in ѕ U 30 so that the case e = 30 is, in a sense, the first non-trivial case to be considered.
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