1 Introduction
The notion of transitivity in the context of Abelian group and module theory was introduced by Kaplansky in [10] and developed further in [11, Section 18]; in this work, we shall focus exclusively on groups.
The basic idea is simple: given an Abelian group G and elements x,y∈G, G is said to be transitive if, when a certain necessary condition for the existence of an automorphism of G mapping x↦y is satisfied, then there really is an automorphism of G mapping x↦y.
In Kaplansky’s initial work, a key role was played by the Ulm or height sequence of elements.
Recall that if G is an Abelian p-group which is reduced (i.e., does not contain a divisible subgroup), then the height in G, hG(x), of an element x is the ordinal α if x∈pαG∖pα+1G, with the usual convention that hG(0)=∞.
The Ulm or height sequence of x with respect to G is the sequence of ordinals or symbols ∞ given by UG(x)=(hG(x),hG(px),hG(p2x),…); the collection of such sequences is partially ordered pointwise.
Thus, in Kaplansky’s original work, a reduced p-group G was said to be transitive if, for each pair of elements x,y∈G with UG(x)=UG(y), there is an automorphism ϕ of G with ϕ(x)=y.
In terms of p-groups, Kaplansky’s original theorem – see [11, Theorem 24] – was the following.
Theorem (Kaplansky).
If M is a reduced Abelian group with the property that any two elements of M can be embedded in a countable direct summand of M, then M is transitive.
Kaplansky observed that both countable p-groups and groups with no elements of infinite height – the latter are usually called separable p-groups – are transitive.
Interestingly, his proof was simply to cite an exercise given after his celebrated proof of Ulm’s theorem [11, Theorem 14].
Since Kaplansky’s early work, there has been significant attention paid to the notion of transitivity; see, for example, the early contributions of Corner [1], Griffith [7] and Hill [8].
The culmination of the work in this area of the last two authors was a proof that the so-called totally projective groups, a large class of Abelian p-groups which contains all direct sums of countable groups, are transitive.
The necessary condition used by Kaplansky is rather particular to local situations such as Abelian p-groups, but another more general approach, which can be used in the category of all groups, has been considered by Strüngmann and the first author in [5, 6].
There, a group G is said to be weakly transitive if there exists an automorphism of G mapping the element x to the element y when there exist endomorphisms θ,ϕ of G with θ(x)=y and ϕ(y)=x.
Both classical and weak transitivity are examples of phenomena which are widespread; in some not too precisely defined way, most groups have these properties.
Furthermore, thanks to an observation of Corner [1], we know that these “generic” types of transitivity are determined by the action of the endomorphism ring on the first Ulm subgroup, pωG, of the group G in question; in particular, all separable p-groups possess both properties.
In this work, we wish to consider another type of transitivity which is in some sense the polar opposite of the classical notion: not “too many” groups will have this property, but sufficiently many do possess it to make the class interesting.
Definition 1.
A p-group G is said to be transitive with respect to cyclic subgroups if, when X,Y are cyclic subgroups of G with (i) X≅Y, (ii) G/X≅G/Y, there exists an automorphism ϕ of G with ϕ(X)=Y.
We use the abbreviation G is CS-transitive for the full statement “G is transitive with respect to cyclic subgroups”.
Remark 2.
The notion of transitivity with respect to cyclic subgroups also makes sense in the category of all groups; condition (ii) in the definition above is replaced by
X,Y are subgroups of G of the same finite index.
Thus the notion of a CS-transitive non-Abelian group leads to a larger class than the so-called order-transitive p-groups, i.e., the p-groups where, for each positive integer r, the elements of order pr are transitively permuted by the automorphism group.
We shall show later that all finite Abelian p-groups are CS-transitive, but for non-Abelian groups, the concept is very strong; for example, a simple calculation shows that the dihedral group of order 8, D4=〈s,t∣s4=t2=1,tst=s3〉, is not transitive with respect to cyclic subgroups since the elements s2 and t generate isomorphic subgroups of order 2 and hence both are subgroups of finite index 4, but no automorphism of D4 can map t↦s2.
On the other hand, the quaternion group Q8=〈a,b∣a4=1,a2=b2,bab=a〉 has a single cyclic subgroup of order 2 and three subgroups of order 4 (and hence of index 2).
However, it is well known that the automorphism group of Q8 acts transitively on the elements of order 4, and hence Q8 is CS-transitive.
In this paper, we shall not investigate CS-transitivity any further for non-Abelian groups.
Consequently, in the sequel, the word “group” shall mean an additively written Abelian p-group, and we shall further assume that all such groups are reduced, i.e., the Prüfer quasi-cyclic group ℤ(p∞) is not a subgroup.
It is apparent that the notion of being transitive with respect to cyclic submodules is connected in some sense to the notion of the submodules being equivalent; however, we emphasise that we are not taking either concept of isomorphism in the above definition as being understood in the context of valuated groups.
For an excellent discussion of that approach, see the paper [9] by Hill and West.
There is another reason why the notion of CS-transitivity is interesting.
In his celebrated paper [8], Hill introduced the notion of a potentially transitive group: a reduced p-group G is said to be potentially transitive if, given any pair of elements x,y of G with UG(x)=UG(y), there is an isomorphism G/〈x〉≅G/〈y〉.
Clearly, a transitive group is potentially transitive, and it is straightforward to show that if G is potentially transitive and CS-transitive, then it must in fact be transitive.
There are situations in which condition (i) that X≅Y of Definition 1 may be dropped; for example, if G is finite, then condition (i) is an immediate consequence of condition (ii).
More generally, as we shall establish later, the same is true if G is an arbitrary homocyclic p-group.
But condition (i) does not, in general, follow from condition (ii), and we find it convenient to make a formal distinction.
Definition 3.
A p-group M is said to be quotient-transitive if, given any pair of non-zero elements x,y∈M, with M/〈x〉≅M/〈y〉, there is an automorphism ϕ of M with ϕ(x)=y.
It is, however, immediate that, for finite p-groups, the notions of quotient-transitivity and CS-transitivity coincide.
Furthermore, a quotient-transitive group is always transitive with respect to cyclic subgroups.
However, the concepts are different as we shall show shortly.
Remark 4.
Definitions 1, 3 can also be extended to modules in a natural way, but in this paper, we will restrict our attention to Abelian p-groups; in a subsequent work, we will develop the more general notion, particularly in the context of p-adic modules.
We finish this introduction by giving a short summary of our principal results and some comments about notation and terminology.
A first objective of the paper is to establish (in Theorem 2) that finite p-groups are quotient-transitive and hence CS-transitive.
Rather surprisingly, we could find no reference to such a result in the existing literature.
We also establish that there are, however, many situations where infinite p-groups are also quotient-transitive, and in Theorem 8, we obtain a classification of separable quotient-transitive p-groups; such a group is either semi-standard – see the paragraph below – or of the form F⊕⊕λℤ(pn), where λ is arbitrary and F is a finite p-group of exponent less than n.
In Section 4, we illustrate by means of examples the difficulties which arise in relation to the classification of CS-transitive p-groups, even when one restricts attention to direct sums of cyclic groups.
Our notation and terminology are largely standard and agree with those used in the texts of Fuchs [2, 3, 4] and Kaplansky [11].
Given a positive integer n, we say that G is homocyclic of exponent n if G has the form ⊕κℤ(pn) for some (possibly infinite) cardinal κ; we use the notation e(G) to denote the exponent of a bounded p-group G, i.e., the least integer n such that pnG=0.
Of particular importance for this work is a notion named semi-standard by Corner: a p-group G is said to be semi-standard if, for a basic subgroup B of G, there is a decomposition B=B1⊕⋯⊕Bn⊕⋯, where each Bn is homocyclic of exponent n and of finite rank, i.e., Bn is of the form ⊕rnℤ(pn) with rn finite or zero.
This is, of course, equivalent to the finiteness of the Ulm invariants fn(G), where 0≤n<ω.
2 Elementary results
In this short section, we present some simple results that are useful for later discussions.
Example 1.
Suppose that G=A⊕B, where A is an elementary p-group of infinite rank and B=〈b〉 is cyclic of order p2.
Then G is CS-transitive but not quotient-transitive.
Proof.
Let x be an arbitrary element of G so that x can be expressed in the form x=ra+sb for some integers r,s and a∈A.
By absorbing suitable units and utilising standard properties of elementary groups, we may assume that 〈a〉 is a summand of A and that r∈{0,1} and s∈{0,1,p}.
We determine the structure of G/〈x〉 for the various cases.
Consider the situation where x has order p.
Then x has the form x=a, a+pb or pb.
Clearly, G/〈a〉≅G and G/〈pb〉≅A.
In the remaining case, G/〈a+pb〉 is easily shown to be isomorphic to G since we have
G/〈a+pb〉=(〈a+pb〉+〈b〉/〈a+pb〉)⊕A1≅〈b〉/(〈a+pb〉∩〈b〉)⊕A1,
where A=〈a〉⊕A1, and the result follows since 〈a+pb〉∩〈b〉=0.
Suppose now that x has order p2 so that x=b or a+b.
Clearly, G/〈b〉≅A, while if x=a+b, we have as before
G/〈a+b〉=A1⊕(〈a+b〉+〈b〉)/〈a+b〉≅A1⊕〈b〉/(〈a+b〉∩〈b〉)≅A,
the last isomorphism coming from the fact that 〈a+b〉∩〈b〉=〈pb〉.
Now suppose that x,y∈G satisfy o(x)=o(y) and G/〈x〉≅G/〈y〉.
If x (and hence y) has order p, then we must have that {x,y}={a,a+pb}, and as both a, a+pb have the same Ulm sequence and G is a direct sum of cyclic groups, it follows from Kaplansky’s theorem or by direct construction that there is an automorphism ϕ of G interchanging x and y.
On the other hand, if o(x)=p2, then {x,y}={b,a+b}, and again the existence of the required automorphism comes from the fact that b and a+b have the same Ulm sequence.
So G is certainly CS-transitive; it is not quotient-transitive since we have that
G/〈b〉≅A≅G/〈pb〉
but clearly no automorphism of G can map b↦pb.
∎
There are, however, situations other finiteness in which quotient-transitivity and CS-transitivity coincide.
Proposition 2.
If G is a finite p-group or a homocyclic p-group of arbitrary rank, then G is CS-transitive if and only if G is quotient-transitive.
Proof.
We have already observed the equivalence of the two transitivity notions when G is finite.
If G is quotient-transitive, then certainly it is CS-transitive.
Suppose that G is homocyclic of exponent n and G is CS-transitive.
If X,Y are cyclic subgroups of G with G/X≅G/Y, then we can find elements x,y∈G with X=〈prx〉, Y=〈psy〉 and x,y are both elements of order pn.
Consequently, 〈x〉 and 〈y〉 are summands of G, say G=〈x〉⊕G1=〈y〉⊕G2, and note that G1≅G2.
Now we have
ℤ(pr)⊕G1≅G/X≅G/Y≅ℤ(ps)⊕G2.
It follows immediately that r=s and so X≅Y.
Thus there is an automorphism ϕ of G with ϕ(x)=y, and G is quotient-transitive, as required.
∎
A simple adaptation of the argument above yields the following.
Corollary 3.
If G is an arbitrary homocyclic p-group, then G is transitive with respect to cyclic subgroups.
In particular, arbitrary homocyclic p-groups are also quotient-transitive.
3 Torsion groups
Our objective in this section is to show that finite p-groups are quotient-transitive.
The corresponding result for transitivity in the original sense of Kaplansky is reasonably straightforward, although, as mentioned in the introduction, it is based on quite a deep result; surprisingly, the argument to establish quotient-transitivity is quite complex.
Clearly, a first step in this process is an understanding of the structural significance of a quotient of the type G/〈x〉.
Fortunately, the quotients that we shall need to handle are of a very specific type, and this eases the calculations required to understand them.
Suppose that G is a p-group and x∈G has Ulm sequence
UG(x)={r0,…,rn=∞},
where the ri (i<n) are integers.
If rn1,rn2…rnt=rn denote the entries following a gap in the Ulm sequence, then by a well-known result [3, Lemma 65.4] or [4, Chapter 10, Lemma 1.4] (attributed by Fuchs to Baer), there is a finite direct summand
C=〈c1〉⊕〈c2〉⊕⋯⊕〈ct〉
of G, where o(ci)=pni+ki, x=pk1c1+pk2c2+…+pktct and the following relationships hold:
k1=r0,k2=rn1-n1,…,kt=rnt-1-nt-1;
We remark that, in the following proposition, we interpret the group ℤ(p0) as the trivial group.
Proposition 1.
Suppose that C=〈c1〉⊕〈c2〉⊕⋯⊕〈ct〉 is a p-group and o(ci)=pni+ki for each 1≤i≤t.
If x=pk1c1+pk2c2+…+pktct, with 0<n1<…<nt and 0≤k1<…<kt, then the quotient group
C/〈x〉≅ℤ(pr0)⊕ℤ(prn1)⊕⋯⊕ℤ(prnt-1),
where r0=k1,rn1=k2+n1,…,rnt-1=kt+nt-1.
In particular, C and C/〈x〉 do not have any cyclic summands of the same order.
Proof.
Notice firstly that the properties of the ni,ki and the form of the element x ensure that UC(x)={r0,r1,…,rn=∞} and that the rn1,rn2…rnt=rn are precisely the entries following a gap in the Ulm sequence so that the relationships in (i) above are satisfied; equivalently the relationships in the final part of the statement of the proposition hold.
The proof is by induction on the number t of summands in the decomposition of C.
If t=1, the result holds since C/〈x〉 is then just cyclic of order pk1 and k1=r0.
So suppose C has t>1 summands and the result is true for groups of the same form with less than t summands.
For convenience, we write X=〈x〉 and A=〈ct〉+X.
If
c=v1c1+v2c2+…+vtct
(for suitable integers vi) is an arbitrary element of C, observe that
pnt-1+ktc=pnt-1+ktvtct
since the conditions on the ni,ki ensure pnt-1+kt≥o(ci) for all 1≤i≤t-1.
Furthermore, we have pnt-1x=pnt-1+ktct, so pnt-1+ktc∈X for all c∈C.
Hence C/X has exponent at most nt-1+kt.
Notice, however, that pnt-1+kt-1ct∉X, for otherwise, it would follow that
pnt-1+kt-1-1ct-1=pnt-1-1x-pnt-1+kt-1ct∈X,
and a simple linear independence argument shows that this cannot be true.
Thus C/X has exponent exactly nt-1+kt=rnt-1.
It also follows from this argument that A/X=〈ct+X〉 is a cyclic subgroup of C/X with exponent equal to the exponent of C/X.
Now it is known [11, Lemma 4] that if G is a p-group with prG=0 and x is an element of order pr, then the cyclic summand generated by x is a direct summand of G.
Thus A/X is a direct summand of C/X, say C/X=A/X⊕B/X, of order prnt-1.
Let C′=〈c1〉⊕⋯⊕〈ct-1〉 so that
B/X≅C/A=(〈ct〉+X+C′)/A≅C′/(C′∩A).
Consider the intersection C′∩A.
Clearly,
〈pk1c1+…+pkt-1ct-1〉=〈x-pktct〉∈C′∩A;
however, if z∈C′∩A, then z=uct+vx=w1c1+…wt-1ct-1, and linear independence gives (w1-vp1k)c1=0,…,(wt-1-vpkt-1)ct-1=0, and so
z=w1c1+…wt-1ct-1=vd,
where d=pk1c1+…+pkt-1ct-1=x-pktct.
Thus we have z∈〈d〉 and so C′∩A=〈d〉.
It follows that B/X≅C′/〈d〉, and since the element d of C′ satisfies the requirements in the statement of the proposition, a simple induction gives that B/X is a direct sum of cyclic groups with exponents equal to the terms following the jumps in the Ulm sequence in C′ of d.
However, it is immediate that these cyclic groups are then of exponents r0,r1,…,rnt-2 and C/〈x〉 has the desired form.
Finally, the conditions on the ni,kj ensure that rni=ni+ki+1 can never be equal to nj+kj for any j.
∎
Summarising Proposition 1, we see that C/〈x〉 determines the Ulm sequence of x in C; the terms following the jumps in UC(x) can be read off from the quotient, and the remaining terms in the Ulm sequence just increase by 1 at each step, so the full sequence can be reconstructed uniquely from knowledge of the quotient C/〈x〉.
We can now use this to obtain the main result of this section.
Theorem 2.
If G is a finite p-group and G/〈x〉≅G/〈y〉 for elements x,y∈G, then UG(x)=UG(y) and there is an automorphism ϕ of G with ϕ(x)=y.
Proof.
Notice immediately that, from the isomorphism between the quotients, it must follow that x,y have the same order.
Suppose that
UG(x)=(r0,…,↑rn1,…,↑rnt=∞),
UG(y)=(s0,…,↑sm1,…,↑smu=∞),
where ↑ indicates a gap in the Ulm sequence; note then that nt=mu.
Embed x,y in summands C,D as in Proposition 1, and let
G=C⊕H=D⊕K;
thus C is a direct sum of cyclic groups of the form C=〈c1〉⊕⋯⊕〈ct〉 with o(ci)=pni+ki, where 0<n1<…Nt and 0≤k1<k2<…<kt; similarly, D is a direct sum of cyclic groups 〈d1〉⊕⋯⊕〈du〉 with o(dj)=pmj+ℓj.
Note that the summands of C,D each consist of a single copy of the given cyclic group so that the Ulm invariants of C,D will be either 1 or 0.
Now G/〈x〉=C/〈x〉⊕H≅G/〈y〉=D/〈y〉⊕K, and we also have, for all α, the relations
fα(G)=fα(C)+fα(H)=fα(D)+fα(K),
where the fα are Ulm invariants.
Furthermore,
fα(C/〈x〉)+fα(H)=fα(D/〈y〉)+fα(K).
Since all the Ulm invariants are integers, we can write fα(H)=fα(G)-fα(C) and a similar expression for fα(K), from which we deduce that
(3.1)fα(C/〈x〉)+fα(D)=fα(D/〈y〉)+fα(C).
Let α=n1+k1-1 so that fα(C)=1, and hence it follows from Proposition 1 that fα(C/〈x〉)=0.
Substituting in (3.1), fα(D)=fα(D/〈y〉)+1, and we conclude that fα(D)≠0.
In particular, D has a summand isomorphic to ℤ(pn1+k1).
Repeating this argument for the various values of α=ni+ki-1, we deduce that D has a subgroup isomorphic to C.
Now choose α to take the various values mj+ℓj, and repeat the argument above.
From this, we can conclude that C has a subgroup isomorphic to D; whence D≅C.
Since the group G is finite, we must have H≅K and C/〈x〉≅D/〈y〉.
An isomorphism ϕ between D and C maps D/〈y〉 isomorphically onto C/〈ϕ(y)〉, and so we conclude from Proposition 1 that UC(x)=UC(ϕ(y)).
Since C is a summand of G and ϕ is an isomorphism, we then have UG(x)=UG(y).
The existence of an automorphism ϕ of G mapping x↦y follows from Kaplansky’s theorem.
∎
3.1 Infinite torsion groups
Our first result in this subsection guarantees the existence of an adequate supply of automorphisms in certain situations; our arguments are based on those of Pierce in [12, Lemma 2.4].
Lemma 3.
Let G be an arbitrary unbounded reduced p-group which is semi-standard and x,y∈G are such that
Then there is an automorphism of G mapping y↦x.
Proof.
To see this, note that, since (i) holds, we can find finite subgroups C1,C2 with x∈C1≤B1, where B1 is a basic subgroup of G, similarly for y∈C2≤B2.
Furthermore, C1,C2 are summands of G, say G=C1⊕E1=C2⊕E2 for suitable E1,E2.
In fact, since B1≅B2 are both direct sums of cyclic groups, we can arrange that C1≅C2, via ψ say.
As the groups C1≅C2 are finite, it follows from standard cancellation theory that we also have E1≅E2; let σ be an isomorphism E1≅E2.
Then we have
C1/〈x〉⊕E1=G/〈x〉≅G/〈y〉=C2/〈y〉⊕E2.
Since G is semi-standard, each Ulm invariant fn(G) is finite, and hence each invariant fn(E1)=fn(E2) is also finite.
The finiteness of the quotients C1/〈x〉 and C2/〈y〉 then yields that C1/〈x〉≅C2/〈y〉.
It follows that C2/〈ψ(x)〉≅C1/〈x〉≅C2/〈y〉, and so, by Theorem 2 above, there is an automorphism θ of C2 with θ(y)=ψ(x).
Now define χ:G→G by χ=ψ-1θ⊕σ, and note that χ is then an automorphism of G with
χ(y)=ψ-1θ(y)=x.∎
In particular, we have established the following result.
Corollary 4.
Let G be a reduced semi-standard separable p-group.
Then G is quotient-transitive.
Proof.
Since G is separable, condition (i) of Lemma 3 holds for all x∈G, and so the result follows if G is unbounded.
If G is bounded and semi-standard, then it is finite, and the result follows from Theorem 2.
∎
The necessity of the condition that C be semi-standard follows from Example 1 in Section 2.
4 Quotient-transitive p-groups
In this section, we obtain a complete classification of separable torsion quotient-transitive p-groups.
First we establish a general result.
Proposition 1.
Suppose that G is quotient-transitive [CS-transitive] and H is a summand of G which is weakly transitive.
Then H is also quotient-transitive [CS-transitive].
Proof.
We give a proof for quotient-transitivity; the simple modification to obtain the corresponding result for CS-transitive groups is left to the reader.
So suppose that G is quotient-transitive and H is a weakly transitive summand of G, say G=H⊕A.
Now if x,y are non-zero elements of H with H/〈x〉≅H/〈y〉, then z=(x,0),w=(y,0) are non-zero elements of G with G/〈z〉≅G/〈w〉, and so there exists an automorphism ϕ of G mapping z↦w; the inverse of ϕ then maps w↦z.
So there is a matrix (αγδβ) sending z↦w.
It follows that α(x)=y for the endomorphism α∈End(H).
Similarly, using the inverse of ϕ, we get an endomorphism α1 of H with α1(y)=x.
Since H is weakly transitive, there is an automorphism of H mapping x↦y, and H is quotient-transitive, as required.
∎
Our next example is elementary but useful; it is a simple extension of part of Example 1.
Example 2.
If G=A⊕B, where A=⊕λℤ(pn), with λ infinite and B=〈b〉 is cyclic of order pm with m>n, then G is not quotient-transitive.
Proof.
As m>n≥1,pnb≠0, and so
G/〈pnb〉≅A⊕ℤ(pn)≅A≅G/〈b〉
since λ is infinite.
Clearly, no automorphism of G can map b↦pnb, and G is not quotient-transitive.
∎
An immediate consequence of Example 2 is the following.
Proposition 3.
If G is quotient-transitive and the Ulm invariant fG(n-1) is infinite for some integer n≥1, then fG(m)=0 for all integers m≥n.
Proof.
Since fG(n-1) is infinite, G has a summand of the form ⊕λℤ(pn) with λ infinite.
If now fG(m)≠0 for any m≥n, then G also has a summand H of the form ⊕λℤ(pn)⊕ℤ(pm).
Since the group H is clearly weakly transitive – it is even transitive – it follows from Proposition 1 that H is quotient-transitive contrary to Example 2.
Thus fG(m)=0 for all m≥n.
∎
It is now easy to deduce a useful finiteness type result.
Proposition 4.
If G is a quotient-transitive p-group and fG(n-1) is infinite for some integer n≥1, then G is pn-bounded.
Proof.
From Proposition 3, we have that fG(m)=0 for all integers m≥n, which implies that a basic subgroup of G is pn-bounded; whence G itself is pn-bounded.
∎
Corollary 5.
If G is a quotient-transitive p-group, either G is semi-standard or G is of the form G=F⊕⊕λZ(pn), where λ is infinite and F is finite of exponent less than n.
Before proceeding to the classification of separable quotient-transitive p-groups, we need a further result.
Proposition 6.
Suppose that G=F⊕B, where F is finite, B is homocyclic of exponent n and of arbitrary rank, and e(F)<n, then G is quotient-transitive.
Proof.
Let 0≠x,y∈G with G/〈x〉≅G/〈y〉.
Then
x=f+pkb,y=f1+pub1
for some integers k,u≥0 and elements b,b1 of height zero in B.
Since b,b1 generate summands of B, we may without loss in generality assume b=b1.
So x=pkb, y=pub; let B=〈b〉⊕B1, and set X=〈x〉, Y=〈y〉.
Now G/〈x〉=((F⊕〈b〉)/〈x〉)⊕B1≅G/〈y〉=((F⊕〈b〉)/〈y〉)⊕B1.
If b+X and b+Y both have order <n, it follows that (F⊕〈b〉/〈x〉) and (F⊕〈b〉/〈y〉) are both of exponent less than n, then by comparing Ulm invariants, one sees immediately that (F⊕〈b〉/〈x〉)≅(F⊕〈b〉/〈y〉).
Since F⊕〈b〉 is finite, it follows from Theorem 2 that there is an automorphism of F⊕〈b〉, and hence of G, mapping x↦y.
The next possibility is that both b+X, b+Y have order pn.
It follows from Lemma 7 below that
F⊕〈b〉/X=ℤ(pn)⊕W,
while F⊕〈b〉/Y=ℤ(pn)⊕V; here W≅F/〈f〉, V≅F/〈f1〉, and hence both are finite of order at most |F|.
Since W,V have exponents at most e(F)<n, it again follows from G/X≅G/Y that W≅V.
Applying Theorem 2 again, we get an automorphism of G mapping x↦y.
The remaining possibility is that one of b+X, b+Y has order pn, while the other has order <pn, say o(b+X)=pn,o(b+Y)<pn; we show that this case cannot arise.
It follows from Lemma 7 below that
F⊕〈b〉/X=ℤ(pn)⊕W,
with W≅F/〈f〉, while F⊕〈b〉/Y is finite of exponent less than n.
So we have G/X≅ℤ(pn)⊕W⊕B1≅(F⊕〈b〉/Y)⊕B1.
Comparison of Ulm invariants again gives
F/〈f〉≅W≅(F⊕〈b〉/Y).
If o(Y)<pn, then in the displayed equation above, the left-hand side has order at most |F|, while the right-hand side has order greater than |F| – a contradiction.
So we conclude that |Y|=o(y)=pn.
But then we must have |F/〈f〉|=|F| which forces f=0.
Hence pkb=x, and since, by assumption, o(b+X)=pn, we are forced to conclude that k≥n; whence x=0 – a contradiction.
So this case does not arise.
Since we have established the existence of an automorphism of G mapping x↦y in the other two cases, the proof will be complete once we have established Lemma 7.
∎
Lemma 7.
Let H=F⊕〈b〉, where b is of order pn and F is finite of exponent strictly less than n.
If a=f+pkb∈H and A=〈a〉, then either H/A is finite of exponent less than n, or H/A≅Z(pn)⊕(F/〈f〉).
Proof.
The first possibility arises if o(b+A)<pn since, in this situation, all the generators of H/A have order less than pn.
If, however, o(b+A)=pn, then the group generated by b+A has order pn and hence is a summand of H by [11, Lemma 4].
Thus H/A splits as (〈b〉+A/A)⊕W, and clearly,
W≅(F⊕〈b〉)/(〈b〉+A)=(F⊕〈b〉)/(〈f〉⊕〈b〉)≅F/〈f〉.∎
The classification we sought is then the following.
Theorem 8.
A separable p-group G is quotient-transitive if and only if it is either semi-standard or of the form G=F⊕⊕λZ(pn), where λ is arbitrary and F is finite of exponent less than n.
Proof.
The necessity has already been established in Corollary 5.
For the sufficiency, we simply quote Proposition 4 and Proposition 6.
∎
We finish this section with a short discussion of the problem of classifying reduced separable CS-transitive p-groups.
In fact, such a classification seems rather difficult even in the simplified situation where we consider only direct sums of cyclic groups.
We begin with a simple general result.
Proposition 9.
If G is a p-group and m≥0 is an integer such that the Ulm invariants fG(m) and fG(2m+1) are both infinite, then G is not CS-transitive.
In particular, the group ⊕λZ(p)⊕⊕μZ(p2) is not CS-transitive if λ,μ are both infinite.
Proof.
If both Ulm invariants are infinite, then G has a decomposition of the form G=A⊕B⊕H, where A=⊕λℤ(pm+1),B=⊕μℤ(p2m+2) and λ,μ are infinite.
Suppose that 〈x〉,〈z〉 are canonical summands of A,B respectively, and let y=pm+1z.
Then o(x)=o(y), and G/〈x〉≅G since λ is infinite.
Furthermore, if B=〈z〉⊕B1, we have G/〈y〉≅A⊕H⊕B1⊕(〈z〉/〈pm+1z〉)≅G.
Since the heights of x and y differ, no automorphism of G can map x↦y, and thus G is not CS-transitive.
∎
Thus it is not true that arbitrary direct sums of cyclic p-groups are transitive on cyclic subgroups; this is in marked contrast to the situation pertaining in classical transitivity, where Kaplansky’s theorem tells us that all such groups are transitive.
It also indicates that no direct analogue of Corner’s use of endomorphism rings discussed in the introduction can hold.
Our next set of examples illustrates how CS-transitivity of direct sums of cyclic groups depends heavily on arithmetical properties of the non-zero Ulm invariants.
Example 10.
Let G=A⊕B, where A=⊕ω〈ei〉,B=⊕ω〈fj〉 and, for all i,j, o(ei)=p, o(fj)=p3.
Then G is CS-transitive.
Proof.
Let x,y be arbitrary elements of G with G/〈x〉≅G/〈y〉.
Since G has exponent 3, there are just 3 classes of elements to consider.
(i) Elements of order p.
There are essentially only three types here: those of the form e, those of the form e+p2f and those of the form p2f, where e∈A, f∈B.
Since all elements of the form e,e+p2f have Ulm sequence equal to (0,∞…), the classical theory of transitivity ensures that there is an automorphism of G interchanging such elements.
Clearly, then G/〈e〉≅G, and it will suffice to show that, for the remaining case, G/〈p2f〉 is not isomorphic to G.
This is immediate since G/〈p2f〉 has a direct summand isomorphic to ℤ(p2).
(ii) Elements of order p2.
Here the situation again arises where there are possibilities of the form e+pf,pf, with f of order p3.
However, if f is an arbitrary element of B of order p3, then G/〈pf〉≅G.
On the other hand, an easy calculation gives that G/〈e+pf〉≅G⊕ℤ(p2).
So, in this situation, we never have isomorphic quotients to deal with.
(iii) Elements of order p3.
In this case, the only possibilities are that the elements are of the form e+f,f.
However, both types of element will have Ulm sequence (0,1,2,∞,…), and so there is, as above, an automorphism interchanging them.
Thus, in all situations where G/〈x〉≅G/〈y〉, we have an automorphism of G mapping x↦y, and so G is CS-transitive.
∎
Example 11.
Let
G=⊕ω〈ei〉⊕⊕ω〈fi〉⊕⊕ω〈gi〉,
where, for all i<ω,
o(ei)=pr,o(fi)=ps and o(gi)=pt
with r<s<t and r+t=2s.
Then G is not CS-transitive.
Proof.
Take x=prf0, y=psg0 so that o(x)=ps-r=pt-s=o(y).
Note that, as x,y have different heights in G, there cannot be an automorphism of G mapping x↦y.
However, a straightforward calculation shows that G/〈x〉≅G≅G/〈y〉, and so G is not CS-transitive.
∎
