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A basis for the laws of the variety ѕU30

To Bernhard Hermann Neumann on his 60th birthday

Published online by Cambridge University Press:  09 April 2009

C. Christensen
C. Christensen
Affiliation:
Australian National University Canberra
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A group is called an ѕU-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 ѕUe 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 ѕUe 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 [xm, ym]m where e = pαm, 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 ѕUe is clearly soluble whikt PSL(2, 5), the non-abelian simple group of order 60, is contained in ѕU30 so that the case e = 30 is, in a sense, the first non-trivial case to be considered.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 3-4 , November 1969 , pp. 493 - 496
Copyright
Copyright © Australian Mathematical Society 1969

References

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