The Theory of Infinite Soluble Groups

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Robinson, see [3] 5. We give below what presumably are much shorter proofs of these interesting results. In fact we prove the following. The following are equivalent. An equivalent definition, often more convenient, is the following.

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A group G has Hirsch number h if G has a series of finite length with exactly h of the factors infinite cyclic, the remaining factors of the series being locally finite; G satisfies min-p if it satisfies the minimal condition on p-subgroups. It is elementary that to within some normal subgroup of finite index, locally finite factors can essentially be moved down a series past torsion-free abelian factors of finite rank and finite factors can be moved up past torsion-free abelian factors of finite rank and past periodic abelian factors satisfying min-p for all primes p.

Further divisible abelian factors in a periodic FAR group sink to the bottom e. Thus it is elementary to see that a group G is a finite extension of a soluble FAR group if and only if it has a characteristic series. Suppose G is a finite extension of a soluble FAR group with its maximal periodic normal subgroup t G finite. Then from the above, the following hold. Then G is residually a finite n -group if and only if Z1G is n-reduced. The proof of the theorem a implies b.

Let H be a normal subgroup of G that is residually a finite n-group. Then Z1 FittH is n-reduced by Lemma 1. Also t H is a n-group. If P is a p-subgroup of t H then P is Chernikov, residually finite and hence finite, and n is finite. Consequently t H is finite.

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This is actually the core of the proof of the Theorem. By b and Lemma 2 FittH is residually a finite n-group, so N is residually a finite n-group. We claim that M is residually a finite nilpotent n-group. If so then c holds. It follows that M is residually a finite nilpotent n-group. G is residually finite, so G is reduced. Since H is residually finite-n, so H contains no n-divisible elements of infinite order. Consequently neither does G. Then H is torsion-free -by-finite, t H is finite and consequently t G is finite. Further we may choose H torsion-free.

Then H has no non-trivial n-divisible elements by d and hence Z1 FittH is n-reduced. Thus b holds. The proof of the corollary If a holds, then so does b by the Theorem. Clearly b implies c. Suppose c holds. By Lemma 1 there exists a finite set k of primes such that Z1 FittG is k-reduced. Hence by the Theorem, b implies a , there exists a normal subgroup H of G of finite index that is residually a finite k-group. Thus a holds. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

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