Abstract
We present proofs of D. N. Azarov’s recent three theorems determining precisely when a soluble group of finite rank is residually a finite \(\pi \)-group for a specified finite set \(\pi \) of primes. Our proofs seem to be substantially shorter; they also apply to groups with a somewhat weaker notion of finite rank.
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According to its English summary Azarov’s paper [1] is devoted to proving the following. If \(\pi \) is a finite set of primes, then a soluble group of finite rank is a finite extension of a residually a finite \(\pi \)-group if and only if it is a finite extension of a residually finite nilpotent \(\pi \)-group, which happens if and only if it is a reduced FATR group with no \(\pi \)-divisible elements of infinite order (see below or see [3] for definitions). Further he proves that a soluble group of finite rank is residually a finite \(\pi \)-group for some finite set \(\pi \) of primes if and only if it is a reduced FATR group. (This is effectively an old result of D. J. S. Robinson, see [3] 5.3.8 or [4] Page 138.) We give below what presumably are much shorter proofs of these interesting results. In fact we prove the following. (Below \(\tau \)(G) denotes the maximal periodic normal subgroup of a group G, FittG its Fitting subgroup and {\(\zeta _{\alpha }\)(G)} its upper central series.)
Let G be a finite extension of a soluble FAR group and let \(\pi \) be some finite set of primes. The following are equivalent.
-
(a)
G is a finite extension of a residually finite \(\pi \)-group.
-
(b)
\(\tau \)(G) is finite and \(\zeta _{1}\)(FittH) is \(\pi \)-reduced for some normal subgroup H of G of finite index.
-
(c)
G is a finite extension of a residually finite nilpotent \(\pi \)-group.
-
(d)
G is a finite extension of a reduced soluble FATR group with no \(\pi \)-divisible elements of infinite order.
Suppose G is a finite extension of a soluble FAR group. The following are equivalent.
-
(a)
There exists a finite set \(\pi \) of primes such that G is residually a finite \(\pi \)-group.
-
(b)
G is reduced and a finite extension of an FATR group.
-
(c)
\(\tau \)(G) is finite and \(\zeta _{1}\)(FittG) is reduced.
Soluble FAR and FATR groups are defined in [3]. An equivalent definition, often more convenient, is the following. A soluble group G is FAR if it has finite Hirsch number and satisfies min-p for every prime p. (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.g. [2] 3.18). 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
where G\(_{1}\) is periodic, divisible, abelian and satisfies min-p for all primes p, G\(_{i+1}\)/G\(_{i}\) for \(1\le i<r\) is infinite periodic abelian with all its primary components (i.e. its Sylow subgroups) finite, G\(_{i+1}\)/G\(_{i}\) for \(r\le i< \mathrm{s}\) is torsion-free abelian of finite rank and G/G\(_{s}\) is finite. The soluble FATR groups are exactly those G above with G\(_{1}\) involving only finitely many primes, with r \(= 1\) and, of course, with G/G\(_{s}\) soluble.
Suppose G is a finite extension of a soluble FAR group with its maximal periodic normal subgroup \(\tau \)(G) finite. Then from the above, the following hold.
-
(a)
G is (torsion-free)-by-finite and
-
(b)
G is a finite extension of a soluble FATR group. Further
-
(c)
FittG is nilpotent and G/FittG is abelian-by-finite (see [3] 5.2.2) and
-
(d)
G/FittG is a finite extension of a free abelian group of finite rank (see [3] 5.2.3).
Note that in general FittG\(_{s}\) = G\(_{s}\cap \)FittG, FittG/FittG\(_{s}\) is finite and FittG is nilpotent if and only FittG\(_{s}\) is nilpotent. Also the soluble groups G of finite rank discussed by Azarov in [1] are exactly FAR groups above with G/G\(_{s}\) soluble and G\(_{r }\)of finite rank.
1 The proofs
We use the following elementary results.
Lemma 1
Let A be a torsion-free abelian group.
-
(a)
If \(\pi \) is any set of primes then A is residually a finite \(\pi \)-group if and only if A is \(\pi \)-reduced.
-
(b)
If A is reduced and of finite rank, then for some finite set \(\pi \) of primes, A is residually a finite \(\pi \)-group.
For example see [3] 5.3.4 and 5.3.5. We also use the following, see [4] 9.38.
Lemma 2
Let G be a nilpotent FAR group and \(\pi \) any set of primes. Then G is residually a finite \(\pi \)-group if and only if \(\zeta _{1}\)(G) is \(\pi \)-reduced.
Thus in Lemma 2 if \(\zeta _{1}\)(G) is \(\pi \)-reduced, then G is residually a finite \(\pi \)-group, so each G/\(\zeta _{i}\)(G) is also residually a finite \(\pi \)-group by Learner’s Lemma. Hence each \(\zeta _{i+1}\)(G)/\(\zeta _{i}(\)G) is \(\pi \)-reduced, something that is easy to see directly.
The proof of the theorem
(a) implies (b). Let H be a normal subgroup of G that is residually a finite \(\pi \)-group. Then \(\zeta _{1}\)(FittH) is \(\pi \)-reduced by Lemma 1. Also \(\tau \)(H) is a \(\pi \)-group. If P is a p-subgroup of \(\tau \)(H) then P is Chernikov, residually finite and hence finite, and \(\pi \) is finite. Consequently \(\tau \)(H) is finite. Clearly \(\tau \)(G)/\(\tau \)(H) is finite, so \(\tau \)(G) is finite.
(b) implies (c). (This is actually the core of the proof of the Theorem.) There exist normal subgroups N \(\le \) L of G with L \(\le \) H, G/L finite, L/N free abelian of finite rank and N = FittL torsion-free nilpotent of finite rank. Clearly N \(\le \) FittH. By (b) and Lemma 2 FittH is residually a finite \(\pi \)-group, so N is residually a finite \(\pi \)-group.
Set q = \(\Pi _{p\in \pi }\) p and \(M=\cap _i C_L (\zeta _{i+1} ( N)^q\zeta _i ( N)/\zeta _i ( N))\). Clearly N \(\le \) M \(\le \) L and G/M is finite. We claim that M is residually a finite nilpotent \(\pi \)-group. If so then c) holds. Now M/N is free abelian, so M/N at least is residually a finite nilpotent \(\pi \)-group.
Let x \(\in \) N\(\backslash \langle 1\rangle \). Since N is residually a finite \(\pi \)-group, there exists a power m = q\(^{\mu }\) of q with x \(\notin \) N\(^{m}\). Now N/N\(^{m}\) is finite and M/N is polycyclic, so there exists a torsion-free normal subgroup T/N\(^{m}\) in M/N\(^{m}\) with M/T finite. Also N/N\(^{m}\) lies in the hypercentre of M/N\(^{m}\) and M/N is abelian. Hence M/N\(^{m}\) is nilpotent, as therefore is its finite image M/T. Let S/T denote the Hall \(\pi \)’-subgroup of M/T. Then M/S is a finite nilpotent \(\pi \)-group. Further x \(\notin \) S, since x is a non-trivial \(\pi \)-element modulo N\(^{m}\) and S/N\(^{m}\) is an extension of a torsion-free group by a \(\pi \)’-group. It follows that M is residually a finite nilpotent \(\pi \)-group.
(c) implies (a). This is trivial. Thus (a), (b) and (c) are equivalent.
(a) and (b) imply (d). G is residually finite, so G is reduced. Also \(\tau \)(G) is finite, so G is a finite extension of a residually finite-\(\pi \), reduced FATR group H. Since H is residually finite-\(\pi \), so H contains no \(\pi \)-divisible elements of infinite order. Consequently neither does G.
(d) implies (b). By (d) G has a reduced soluble normal FATR subgroup H of finite index. Then H is (torsion-free)-by-finite, \(\tau \)(H) is finite and consequently \(\tau \)(G) is finite. Further we may choose H torsion-free. Then H has no non-trivial \(\pi \)-divisible elements by (d) and hence \(\zeta _{1}\)(FittH) is \(\pi \)-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 \(\kappa \) of primes such that \(\zeta _{1}\)(FittG) is \(\kappa \)-reduced. Hence by the Theorem, (b) implies (a), there exists a normal subgroup H of G of finite index that is residually a finite \(\kappa \)-group. But then G is residually a finite \(\pi \)-group for \(\pi =\kappa \cup \){all prime divisors of (G:H)}. Thus (a) holds.
Remark
In a special but still quite general case there is a slicker but less elementary proof of (b) implies (c), the main implication of the theorem.
With N as in the original proof let N\(_{p}\) denote the finite-p residual of N. Then \(\cap _{p\in \pi } N_p =\langle 1\rangle \). The upper central factors of N/N\(_{p}\) are p-reduced (Lemma 2). If they are actually (torsion-free)-by-(a p’-group)-by-finite, then G/N\(_{p}\) embeds into GL(n, J) for some integer n and J the integers localized at p. Thus G/N\(_{p}\) is a finite extension of a residually finite p-group and consequently G is a finite extension of a residually finite nilpotent \(\pi \)-group.
References
Azarov, D.N.: Some residual properties of soluble groups of finite rank (Russian with English summary). Chebyshevski Sb. 15, 7–18 (2014)
Kegel, O.H., Wehrfritz, B.A.F.: Locally Finite Groups. North-Holland Pub. Co., Amsterdam (1973)
Lennox, J.C., Robinson, D.J.S.: The Theory of Infinite Soluble Groups. Clarendon Press, Oxford (2004)
Robinson, D.J.S.: Finiteness Conditions and Generalized Soluble Groups, vol. 2. Springer, Berlin (1972)
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Wehrfritz, B.A.F. Remarks on Azarov’s work on soluble groups of finite rank. Boll Unione Mat Ital 9, 319–322 (2016). https://doi.org/10.1007/s40574-015-0047-8
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DOI: https://doi.org/10.1007/s40574-015-0047-8