ON FINITE ABELIAN GROUPS REALIZABLE AS MISLIN GENERA

We study the realizability of finite abelian groups as Mislin genera of finitely generated nilpotent groups with finite commutator subgroup. In particular, we give criteria to decide whether a finite abelian group is realizable as the Mislin genus of a direct product of nilpotent groups of a certain specified type. In the case of a positive answer, we also give an effective way of realizing that abelian group as a genus. Further, we obtain some non-realizability results.


Introduction
The (Mislin) genus ( [5]) of a finitely generated nilpotent group N , denoted G(N ), is the set of isomorphism classes of finitely generated nilpotent groups M having, at each prime p, a localization isomorphic with that of N , i.e.M p ∼ = N p for all p.It was shown in [2], [5] that G(N ) may be given the structure of a finite abelian group, with the isomorphism class of N as identity element, if the commutator subgroup [N, N ] is finite.Thus we are led to study the class N 0 of finitely generated nilpotent groups with finite commutator subgroup.
No general method has yet been discovered for calculating G(N ) when N ∈ N 0 .However, in [1], a general method was given if N ∈ N 1 , where N 1 is the following subclass of N 0 .Here we describe N 1 by introducing the short exact sequence T N N F N associated with the nilpotent group N , where T N is the torsion subgroup of N and F N is the torsion-free quotient.Plainly the class N 0 is given by the conditions that T N be finite and F N free abelian of finite rank.
Then the class N 1 ⊆ N 0 is given by the supplementary conditions (1) T N is abelian; (2) T N N F N splits on the right, so that N is the semidirect product for an action ω : F N → Aut T N; (3) ω(F N) lies in the centre of Aut T N.
Moreover, in the presence of (1), condition (3) is equivalent to (3') given ξ ∈ F N, there exists a positive integer u such that the action of ξ on T N is given by ξ • a = ua for all a ∈ T N (here, T N is written additively).
Let t be the height of ker ω in F N, that is, where (Z/t) * is the multiplicative group of units of Z/t.It was further shown how to associate with every unit m of Z/t a group N m in the genus of N such that provides an isomorphism (1.1).Moreover, an algorithm was given for calculating t, knowing the exponent of T N and the positive integers u referred to in (3').
Unfortunately, the class N 1 is very restricted; indeed, it was shown in [4] that, if a group N in N 1 has non-trivial genus, then F N is cyclic.However, in [3], the systematic calculation of G(N ) was extended from N 1 to the class N 2 consisting of direct products of groups in N 1 .It is plain that conditions (1) and (2) for membership of N 1 are inherited by direct products, but, in general, condition (3) is not.Thus the class N 2 is substantially larger than N 1 .Of course, membership of N 0 is inherited by direct products.
The calculation of G(N ), for N in N 2 , is somewhat technical, but, from our point of view in this paper, the salient facts are the following.
and if F N i is not cyclic for some i, then G(N ) is trivial; indeed, we will generalize this result below (see Corollary 2.2).Now assume that F N i is cyclic for all i (1 ≤ i ≤ k), and, in accordance with (1.1), suppose Notice that G(N ) is entirely determined by the two invariants (t, P ).It is not difficult to show that, given (t, P ) with P ⊆ T t , there is always a group N in N 2 yielding the invariants (t, P ) -see Section 4. We remark that if k = 1, so that N ∈ N 1 , then P is empty.For full details see [3,Theorem 1.6].
Our principal aim in this paper is to describe those finite abelian groups which arise as described in Theorem 1.1 and which can therefore be realized as a (Mislin) genus group G(N ), for some nilpotent group N in N 2 .We will thereby also obtain some non-realizability results.
In Section 2 we obtain some preliminary results which are of independent interest.We adopt the convention that Z/n is written C n when thought of as a multiplicative group.

Preliminary results
As in Section 1, let

1) is a surjective homomorphism.
Proof: In accordance with (1.1) we have the short exact sequence that can be embedded in the commutative diagram where H is the subgroup factored out of (Z/t) * to yield G(N ) and α(m) = m mod t.Then (2.2) may be completed by a homomorphism Ψ i : G(N i ) −→ G(N ) which will be surjective since α is surjective.It remains to show that Ψ i is given by (2.1).Of course, we may assume that under the isomorphism (1.1) between G(N i ) and (Z/t i ) * /{±1}.We may, and shall, choose m from its residue class mod t i to be prime to the order of T N.
Let T be the set of primes p such that T N has p-torsion.
The homomorphisms φ i , ψ i determine, in an obvious way, homomorphisms and ψ is a T -automorphism with det ψ = m.Thus, in the bottom row of (2.2), [m] ∈ (Z/t) * goes to M in G(N ), completing the proof.
From the explicit description in [3] of the set of primes P which appears in our statement of Theorem 1.1 the following conclusions are plain.

Proposition 2.3. Any finite abelian group realizable as
This will simplify our choice of examples in Section 3.
) by taking P to be T t itself.In particular, G(N ×N j ) is independent of j.

Realizing an abelian group as a Mislin genus
We first enunciate two relevant lemmas on finite abelian groups.For these lemmas we will adopt additive notation; and p will always denote a prime. where and the new relation is and we set Our second lemma is very elementary; the proof will be omitted.Both these lemmas will be applied with p = 2.We now apply Theorem 1.1 to prove our main theorem.We denote the Euler totient function by Φ.
Theorem 3.3.The finite abelian groups which are realizable as the genus of a group in N 2 are precisely the groups of the form , where ≥ 0, i ≥ 1, j ≥ 1 and P , Q are disjoint (finite) sets of odd primes.
Proof: We will prove that the finite abelian groups which are realizable as the genus of a group in N 2 are precisely those groups which, in multiplicative notation, are obtained through the following process: Step 1: Take , where the p i are distinct odd primes and i ≥ 1.
Step 2: Reduce the order of µ of the factors Step 3: Take the direct product of the result of Step 2 with C 2 , ≥ 0.
We recall from Theorem 1.1 that N = N 1 ×• • •×N k determines a certain natural number t and that G(N ) is obtained from (Z/t) * by factoring out the residue class −1 and residue classes m such that m ≡ ±1 mod p i i for p i ∈ P , where P is a certain subset (perhaps empty) of T t , the set of prime divisors of t = λ i=1 p i i , and m ≡ 1 mod p i i for p i ∈ T t − P .Obviously, this is equivalent to factoring out −1 and the residue classes m i , where p i ∈ P and (3.1) Assume first that t is odd, so that each p i is odd.Then (Z/t) * is given by Step 1. Factoring out m i simply reduces , where p j is chosen among the primes in T t − P to be such that the 2-valuation of p j − 1 is minimal.If P = T t , then this last part of Step 2 is void (because then −1 = pi∈Tt m i ).
Step 3 is also void if t is odd, that is, we take = 0. Assume now that t is even.Notice that if t = 2t , with t odd, then (Z/t) * ∼ = (Z/t ) * and the process proceeds just as above with (Z/t ) * , using the same subset P and ignoring the prime 2. Thus we may assume that 4 | t; and we change notation to write To pass to G(N ), we first factor out the m i defined as in ( .We see, conversely, that every group achieved by executing the three steps is realizable as G(N ) with N ∈ N 2 -but certainly not uniquely.There is not even always a unique pair (t, P ) giving rise to a given finite abelian group.However, if the group we want to realize is , where P , Q are disjoint finite sets of odd primes, then we realize A by the pair (t, P ), where and 2 / ∈ P (of course, P or Q may be empty).This completes the proof.
We close this section with two observations supplementary to Theorem 3.3.First we characterize those finite abelian groups which can be realized as G(N ) for N in N 1 .We recall that this is equivalent to characterizing the finite abelian groups which can be realized as G(N ) for N ∈ N 2 with P empty.This provides the proof of the following.Proposition 3.4.The finite abelian groups which are realizable as the genus of a group in N 1 are precisely those groups which, in multiplicative notation, are obtained through the following process: Step 1: Take a group , where the p i are distinct odd primes and i ≥ 1.
Step 2: Either (i) reduce the order of some C Φ(p i i ) to 1 2 Φ(p i i ), where p i is chosen so that the 2-valuation of p i − 1 is minimal; or (ii) take the direct product with C 2 , ≥ 0.
Notice that we may simply stop at Step

Examples and supplementary results
We first give some examples of realizability and non-realizability.32 ) so G ∼ = (Z/t) * /{±1} for t = 144.Of course, other values of t will also serve, e.g.t = 104, 112.It is shown in [1] or [4] how any t may be realized by a group N in N 1 .
Example 4.2.We cannot realize the group Z/5 ⊕ Z/9 as G(N ) for N ∈ N 1 .This follows from the fact that 90 is not a value taken by the Euler totient function Φ.For if Φ(t) = 90, then we easily eliminate t = p, p 2 , p 3 (p odd); but if t = 2 +2 p m ( ≥ 0) or t = mpq (q odd), then 4 | Φ(t).
On the other hand, we can realize Z/5 ⊕ Z/9 as G(N ) for N ∈ N 2 .For if we start with C Φ(11) × C Φ(19) = C 10 × C 18 , we reduce the order of both factors to get C 5 × C 9 and Step 3 is void.This realization amounts to choosing N so that t = 836 and P = {11, 19}.(We will see later how to realize any (t, P ) by a group N in N 2 ).
We next prove a theorem on the realizability of cyclic groups of prime power order.Proof: It is plain that if p = 2, p = 3, or 2p m + 1 is prime, then C p m may even be realized as G(N ) for some N in N 1 .To prove the converse, suppose that C p m is obtained from (Z/t) * by factoring out some elementary abelian 2-subgroup H.We assume henceforth that p = 2. Let t = 2 λ i=1 p i i , where each p i is odd and i ≥ 1.Since |G(N )| is to be odd, it is clear that = 0, 1 or 2 (the case = 1 can be ignored in practice) and that all possible reductions of order must take place.Thus (4.1) If any i ≥ 2 then (4.1) implies that p i = p and 1 2 (p i − 1) = 1, so p = p i = 3.If each i = 1, then each group on the right of (4.1) is a p-group, so there can be only one non-trivial factor, say the ith factor, yielding 1  2 (p i − 1) = p m .Thus 2p m + 1 = p i is prime.

Remarks.
(a) Notice that, in fact, t can only have, at most, two odd prime factors, namely 3 and 2p m + 1.
(b) We find a source of genera which are cyclic 2-groups by taking t to be a Fermat prime.Of course, we may take t to be a product of distinct Fermat primes to yield genera which are non-cyclic 2-groups.
(c) Mendelsohn (see [6]) has proved that there exist infinitely many primes p such that 2 n p is not a value of the Φ-function, for any n ≥ 1.For such primes p, no group of order 2 m p, m > 0, can be realizable.
We close by showing how to realize a pair (t, P ), where P ⊆ T t , by a group N in N 2 .We first take P = ∅ and realize t by a group N in N 1 .The procedure given in [1] or [4]

Lemma 3 . 2 .
Let G = Z/p ⊕ B, where the first summand is generated by a, and let b ∈ B with pb = 0.If we obtain G from G by adding the relation a + b = 0, then G ∼ = B.

Theorem 4 . 3 .
Let p be a prime number and m ≥ 1.Then C p m may be realized as G(N ), N ∈ N 2 , if and only if p = 2, p = 3 or 2p m + 1 is prime.
is as follows.Let t be odd, say, t = p 1 1 . . .p λ λ .Set T N = Z/n, where n = p 1+1 1 . . .p λ +1 λ and let F N = ξ act on T N by ξ • a = ua, where u = 1 + p 1 . . .p λ .If N is the semidirect product for this action, then N is nilpotent and G(N ) = (Z/t) * /{±1}.Now let t be even, say t = 2 p 1 1 . . .p λ λ .Set T N = Z/n, where n = 2 +2 p 1+1 1 . . .p λ +1 λ and let F N = ξ act on T N Let t = gcd(t 1 , . . ., t k ) and let T t = {p 1 , . . ., p λ }, where 3.1) with p i ∈ P .This is achieved by a partial Step 2 of the process, applied to λ i=1 C Φ(p i i ) .If 2 ∈ P , we erase C 2 on the right of (3.2) and then factor out −1 (if P = T t ) just as in the case of t odd, by reducing the order of a suitable C with p j ∈ T t − P .If 2 / ∈ P , then we apply Lemma 3.2, factoring out −1 by effectively erasing C 2 .We are thus left with the direct product of C 2 , ≥ 0, and the result of Step 2 applied to λ i=1 C Φ(p i i ) 1. Our second observation relates to Step 2 in Theorem 3.3.Obviously Step 2 involves factoring out of the group taken in Step 1 an elementary abelian 2-subgroup.An easy extension of Lemma 3.1 establishes Theorem 3.5.If H is any elementary abelian 2-subgroup of the group described in Step 1 of Theorem 3.3, then the quotient of this group by H may be achieved by a suitably chosen Step 2.