**(i)** Stabilizer condition ⇒
**(ii)** Tree condition ⇒
**(iii)** Special subgroup ⇒
**(iv)** Subgroups arising from partitions ⇒ **(i)**.

Let me illustrate this here for the results of §5.
For a subgroup *G*
of Sym(Ω) closed in the function topology --

**(i)** is condition (7) of the paper, saying that the
stabilizer in *G* of every finite set has at least
one infinite orbit in Ω.

**(ii)** is the hypothesis of Lemma 10, saying that
one can make a "tree" of choices of elements
β_{0}, β_{1}, ... of Ω, with
infinitely many β_{i} as
acceptable next choices following any
acceptable initial string β_{0}, ...,
β_{i-1}, such
that a certain fixed infinite string α_{0}, α_{1}, ...
can be mapped by elements of *G* to every one of the resulting
infinite strings β_{0}, β_{1}, ... .

**(iii)** says that, after we adjoin one additional
element *s* to *G* (chosen
to act "in every possible way" as "local"
permutations on a family of uncountably many branches of the
above tree), the resulting group contains elements that carry
{α_{0}, α_{1}, ...} into itself, and induce
the full symmetric group of that set.

**(iv)** says that after adjoining finitely many elements
to *G*,
one gets the full symmetric group Sym(Ω).
(The description "subgroups arising from partitions"
for this condition is more natural
in the contexts of §6 and §7.
Suffice it to say here that the full symmetric group
corresponds to the indiscrete partition on Ω.)

In the paper we work backwards.
So, in 5, we first recall a general version of
**(iii)**⇒**(iv)**, namely (8), cited from [5];
then prove **(ii)**⇒**(iii)** in Lemmas 9
and 10.
In the discussion following the latter lemma, we show
**(i)**⇒**(ii)**, and finally, we state Theorem 11,
and as the proof thereof, tie up the argument by noting that the
results of 3 give **(iv)**⇒**(i)**.

There are points in favor of that backwards development: The results being proved are of most interest when one sees what they will be used for; and that order also makes it comfortable to state and prove them under more general hypotheses than would seem natural if we only looked at them as steps linking onto the conclusions of previous results. But whatever order we give them in, it would have been desirable to make clear the global structure.

The details of conditions **(i)**-**(iv)** and of the proofs
connecting them differ among the
three cases considered in 5, 6 and 7.
The contexts of 6 and 7 involve the additional steps of
proving equivalence of the subgroups associated to the different
partitions in the relevant class;
while in 5, the implication **(ii)**⇒**(iii)** has an
intermediate step, the statement that the result of adjoining
*s* to *G* contains elements that act
by all *local* permutations on {α_{i} }.
Some other differences
are noted in 6 and 7 of the paper.

**Note on Theorem 6.**
That theorem shows the inequivalence of
subgroups in the first 3 of the 4 equivalence classes that
the paper treats, but as noted in the paragraph following the
proof, the method of the theorem cannot distinguish the 4th
equivalence class from the 3rd.
Although the property used in the paper to distinguish those
two classes, namely countability, gives an easy argument, let us
note that a little twist on the method of the theorem will also
distinguish that case.
In the case of countable Ω, the fourth condition
to look at is

Ω^{ℵ0} admits
an ℵ_{0}-uncrowded generalized metric with respect
to which all elements of *G* are bounded.

One can strengthen statement (c) of the theorem to say (in the
countable case) that if there is a finite bound on the cardinalities
of the members of *A*, but infinitely many
members of *A* are non-singletons, then
*G* has the property stated under (c)
but *not* the above property; and a statement (d)
can be added, saying that if all members of *A* are
finite, and almost all are singletons, then
*G* does have the property stated above.

(For a general cardinal κ as in the theorem, the
corresponding property is that is that
Ω^{κ} admit a κ-uncrowded
generalized metric with respect to which all elements of
*G* are bounded.
But to say exactly when this holds is more complicated:
one has to bring in the least λ such that
2^{λ} ≥ κ.)

**A further question.**
In the second sentence of 10 of the paper, we ask about
the structure of the join-semilattice of equivalence classes of
subgroups of Sym(Ω) under the equivalence relation
studied there.
Some very simple questions about this semilattice are:
Is it a chain?
Is the top element at least join-irreducible?
I have incorporated the latter question into the paper
*
Some results on embeddings of algebras, after de Bruijn
and McKenzie,* as Question 9.2:
"Suppose *G* and *H* are subgroups
of Sym[(ω)], which together generate that group.
Must Sym[(ω)] be finitely generated over one
of these subgroups?"