IJMMS 32:5 (2002) 281–284 PII. S0161171202203373 http://ijmms.hindawi.com
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MORPHISMS OF MISLIN GENERA INDUCED BY FINITE NORMAL SUBGROUPS
P. J. HILTON and P. J. WITBOOI Received 20 March 2002
We correct an erroneous statement about induced morphisms of Mislin genera and give the correct statement, even under more general hypotheses.
2000 Mathematics Subject Classification: 20F18, 20E34.
As in [9], we denote the class of all finitely generated groups with finite commutator subgroups byᐄ0, and for anᐄ0-groupH, we letχ(H)be the set of isomorphism classes of groupsKfor whichK×ZH×Z. IfH is anilpotent ᐄ0-group, the Mislin genus (i.e., the genus as defined in [4]) ofHis denoted byᏳ(H). By a result of Warfield [6], we know that ifH is a nilpotentᐄ0-group, then χ(H)=Ᏻ(H). Furthermore, for an ᐄ0-groupH, in [9] it is shown that there is an abelian group structure onχ(H)which coincides with the Hilton-Mislin group structure [3] onᏳ(H)ifHis nilpotent.
In [8, Section 3], it was shown how to define a functionη:χ(H)→χ(H/F)ifHis an infiniteᐄ0-group andF is a finite normal subgroup ofH. It was also shown that the function is not always a homomorphism [8, Example 5.4]. This is in conflict with [2, Theorem 1.3]. In fact there is an error in [2, Theorem 1.1] in that the function α∗:Ᏻ(N)→Ᏻ(N/F)is not always well defined. The counterexample of [9] suggests a way to show explicitly how things may go wrong. (To merely show thatα∗ is not always well defined there are simpler examples, but for a simpler example one may find that there is nevertheless some epimorphisms Ᏻ(N)→Ᏻ(N/F).) We will show that the results of [2, Section 1] remain valid.
In order to ensure that the relationα∗of [2, Section 1] is a well-defined function, we could follow the option of replacing the domainᏳ(N)with a different set, which we briefly describe as follows.
Letᏺ0be the subclass ofᐄ0consisting of all infinite nilpotent groups. For anᏺ0- groupHand a suitable finite groupF, we fix a monomorphismh:F→Hwithh(F)H. Now letKbe a group in the Mislin genus ofH, and letk:F→Kbe any monomorphism withk(F)Kwhich admits, for every primep, an isomorphismf:Kp→Hpfor which f◦kp=hp. We denote the class of all such pairs(K,k)byᏴ0. Ifl:F→Lis another such homomorphism, then we say thatl∼kif there is an isomorphismφ:L→Kfor which φ◦l=k. Then∼is an equivalence relation. LetᏳ(H,h)be thesetᏳ(H,h)=Ᏼ0/∼of all equivalence classes of such endomorphisms. SinceᏳ(H)is finite and since there are only finitely many embeddings ofF intoH, it is easy to prove thatᏳ(H,h)is a finite set. At least then we can follow [2, Theorem 1.1]. The association(K,k)K/k(F) determines a functionα∗:Ᏻ(H,h)Ᏻ(H/h(F)). There is of course the difficulty that
282 P. J. HILTON AND P. J. WITBOOI
the setᏳ(H,h)is not well understood, for example, we do not know whetherᏳ(H,h) has a suitable group structure. Anyway, we are interested inᏳ(H), and we will follow a different option.
We know (see, e.g., [7]) that ifFis a characteristic subgroup of the torsion subgroup THofH, then we do have a homomorphismᏳ(H)→Ᏻ(H/F), in fact, an epimorphism.
In the calculation that leads up to [2, Theorem 3.1], the subgroup kerαof N that is being factored out is, indeed, a characteristic subgroup ofT (see Proposition 7).
Further we note that ˜N is of the formH×(Z2)for some groupH, and then by [7, Corollary 4.2] we have an isomorphismᏳ(H)→Ᏻ(N)˜ . For such a groupHwe have (see [1]) thatᏳ(H)=(Z˜t)∗/{1,−1}. Thus it follows that [2, Theorem 3.1] is valid. In this paper, we will find a more general condition on the pairF Hin order to have a homomorphismᏳ(H)→Ᏻ(H/F), in fact, an epimorphism. Our result in this regard is more general in that we do not require the groupHto be nilpotent.
We recall the following invariant of anᐄ0-group.
Definition1(see [9]). For anᐄ0-groupH, letn1be the exponent of the torsion subgroup TH, let n2 be the exponent of the group Aut(TH), and let n3 be the ex- ponent of the torsion subgroup of the center ofH. We define the natural number n(H)=n1n2n3.
Note that ifH is anᐄ0-group andKis a group for whichK×ZH×Z, thenKis also anᐄ0-group andTKTH, so thatn(K)=n(H). Also note that for such groups HandK, if:H→Kis an embedding then the index[K:(H)]is finite.
Theorem 2. Let H be an infinite ᐄ0-group, and let n=n(H). Let F be a finite subgroup ofH. The following two conditions are equivalent:
(1) given any embeddingφ:H→H such that[H:φ(H)]is relatively prime ton, φ(F)=F;
(2) ifLis any group for whichL×ZH×Z, andβ1andβ2are any two embeddings ofLonto subgroupsK1andK2, respectively, ofH, with both[H:K1]and[H:K2] relatively prime ton, thenβ−11 (F)=β−12 (F).
Proof. Assume that condition (1) holds and suppose that we are givenL,β1, and β2as in (2). ThenFis contained in bothK1andK2. In order to prove (2), it suffices to show that, given any isomorphismβ:K1→K2,β(F)=F. By [9, Theorem 4.2] it follows that there is an embeddingγ:H→K1such that[K1:γ(H)]is relatively prime ton (note thatn(H)=n(K1)). Let:K1→H andδ:K2→H be the inclusions. Then we have embeddings◦γandδ◦β◦γofHintoH. By (1), it follows that◦γ(F)=F and δ◦β◦γ(F)=F. Moreover,(F)=F andδ(F)=F, and consequently we haveβ(F)=F. So we have proved that (1) implies (2).
The converse implication is clear.
Remark 3. Notice that for any infinite ᐄ0-group H and any groupL for which L×ZH×Z, Lis an ᐄ0-group andn(L)=n(H). It is then not hard to see that conditions (1) and (2) ofTheorem 2are equivalent to the following condition:
(3) ifβ1andβ2are any two embeddings ofHonto subgroupsK1andK2, respec- tively, ofL, with[L:K1]and[L:K2]relatively prime ton, thenβ1(F)=β2(F).
MORPHISMS OF MISLIN GENERA INDUCED BY FINITE... 283 We are now able to state and prove a significant result on induced morphisms.
Theorem4. LetH be anᐄ0-group, and letn=n(H). LetF be a finite subgroup ofHwith the property that, given any embeddingφ:H→Hsuch that[H:φ(H)]is relatively prime ton, φ(F)=F. Then, for subgroups K ofH with[H:K]relatively prime ton, the associationKK/Fdefines an epimorphismη:χ(H)→χ(H/F).
Proof. We first note that, by implication,F must be a normal subgroup ofH. By the equivalence of (1) and (2) inTheorem 2, it follows thatηis well defined. The proof is completed in a way similar to the proof of [7, Theorem 2.1] using [9, Proposition 6.1].
For anᐄ0-group H, TH has finite characteristic subgroups [TH,TH] and ZTH to which [7, Theorem 2.1] applies. We point out some other subgroups to which the more generalTheorem 4is applicable.
Theorem5. LetH be an infiniteᐄ0-group. LetF=[H,H]∩TH. ThenH, together withF, satisfies condition (1) ofTheorem 2.
Proof. Letφ:H→Hbe any embedding such that[H:φ(H)]is relatively prime ton. Thenφ[H,H]=[φH,φH] < [H,H]. Alsoφ(TH) < TH. Thusφ(F) < F. SinceF is finite, it follows thatφ(F)=F.
Theorem6. LetHbe an infiniteᐄ0-group. LetF=ZH∩TH. ThenHtogether with F satisfies condition (1) ofTheorem 2.
Proof. Letφ:H→Hbe any embedding such that[H:φ(H)]is relatively prime ton. Thenφcan be extended to an isomorphismψ:H×Zk→H×Zkfor somek∈N (see the proof of [9, Theorem 4.1]). NowZ(H×Zk)=(ZH)×Zk. Since the isomorphism ψpreserves centers and preserves torsion, it follows thatψ(F)=F. Since the induced homomorphismφmapsTHisomorphically ontoTH, it follows thatφ(F)=F.
The following result offers an alternative approach to [2, Theorem 3.1], or to a generalization of it.
Proposition7. Letn∈N, and let T=
x,y,z|x2=y2=z2n=1, [x,y]=zn, [x,z]=1=[y,z]
. (1)
Then the subgroupF= x,y,zn ofT is a characteristic subgroup ofT.
Proof. We note thatF is generated by elements of order 2 and every element of order 2 inT is contained inF. ThereforeF is a characteristic subgroup ofT.
Proposition8. Letn,u∈Nbe such thatuis relatively prime to2n. Lettbe the multiplicative order ofumod 2n, and let˜tbe the multiplicative order ofumodn. Let T andF be the groups ofProposition 7, and letζbe the action ofZonT defined (for a∈Z) by
(a,z)→z(ua), (a,x)→x, (a,y)→y. (2)
284 P. J. HILTON AND P. J. WITBOOI
Then, for the groupH=T ζZ,F Hand we have an epimorphismχ(H)→χ(H/F)= (Z˜t)∗/{1,−1}.
In particular, if˜t=t, thenχ(H)χ(H/F).
Proof. Our conditions ensure that indeedζis an action. ByProposition 7,F is a characteristic subgroup ofT, and thus byTheorem 4, there is an epimorphismχ(H)→ χ(H/F). The groupH/F is isomorphic to the group
a,b|an=1, bab−1=au
(3) and therefore by [5, Theorem 3.8] we haveχ(H/F)=(Z˜t)∗/{1,−1}.
By [8, Theorem 2.6] there is an epimorphism Zt∗
/{1,−1}→χ(H), (4)
and so, if ˜t=t, thenχ(H)χ(H/F).
References
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[3] P. Hilton and G. Mislin,On the genus of a nilpotent group with finite commutator subgroup, Math. Z.146(1976), no. 3, 201–211.
[4] G. Mislin,Nilpotent groups with finite commutator subgroups, Localization in Group The- ory and Homotopy Theory, and Related Topics (Sympos., Battelle Seattle Res. Cen- ter, Seattle, Wash., 1974), Lecture Notes in Math., vol. 418, Springer, Berlin, 1974, pp. 103–120.
[5] D. Scevenels and P. Witbooi,Non-cancellation and Mislin genus of certain groups andH0- spaces, J. Pure Appl. Algebra170(2002), no. 2-3, 309–320.
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Pure Appl. Algebra6(1975), no. 2, 125–132.
[7] P. J. Witbooi,Epimorphisms of non-cancellation groups, in preparation.
[8] ,Non-cancellation for groups with non-abelian torsion, in preparation.
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P. J. Hilton: SUNY at Binghamton, Binghamton, NY13902-6000, USA
Current address:University of Central Florida, Orlando, Florida32816, USA E-mail address:[email protected]
P. J. Witbooi: University of the Western Cape, Private BagX17,7535Bellville, South Africa
E-mail address:[email protected]
