Additional changes: * Thank referee in acknowledgements. * Tweaked grant acknowledgement to better fit EJC style. * Added reference to Shewale/Joshi/Kharat and the more recent Joshi/Waphare/Kavishwar. * Changed tense when referring to Polymath project from "is currently" to "has been", since it is not clear that the FUNC polymath project is still active (last activity in December, last activity of much volume a year or more ago). * Added Remark 1.8, which points out that Shewale/Joshi/Kharat state essentially Corollary 1.7 in a short Remark after their main theorem. > Report on the paper “Frankl’s Conjecture for subgroup lattices” by > Alireza Abdollahi, Russ Woodroofe and Gjergji Zaimi > > This paper is well motivated and the results are interesting. If the authors > slightly improve the exposition according to the suggestions below, I > definitely recommend its publication. > 1. Page 1. Statement of Frankl’s Conjecture. > Definition of join-irreducible: easy to guess, but should be stated > explicitly, because of its relevance. Added a definition in the 2nd sentence. > 2. Page 1, line -8. we will say that L So fixed. > 3. Page 1, line -6. > : namely that the conjecture holds for. . . Added namely. It seemed to us that a comma is also required after the namely, so put that in. > 4. Page 2, Corollary 1.4. > Do the authors mean: If G is a group, with a normal subgroup N ̸= G, such > that G/N is generated by at most two elements, then . . . ? Yes, this wording is an improvement. > 5. Page 3, line 1. > The proof of Theorem 1.2 will be obtained combining Corollary 1.4 with > results on finite simple groups. Ok, followed this wording. Assume "by combining" was meant. > 6. Page 3, Proposition 1.6 > If G is a supersolvable group in which all Sylow subgroup are elementary > abelian Yes, much better. > 7. Page 3, Section 2.1. > More clear for the reader to start: Proof of Corollary 1.4. and then write > the complete proof. (There is no need to give Dedekind Modular identity as a > Lemma. It is well known and based on the elementary implication hn = k =⇒ > n = h−1k ∈ N ∩ K. In any case the proof is given in 1.3.14, page 15 of > the book of D.J.S. Robinson, A course in the theory of groups, > Springer-Verlag. Why quote it as a proposed exercise ?). It is our opinion that, although the Dedekind Identity is well-known to experts in group theory, that people in extremal combinatorics or lattice theory might also be interested in our paper, and might be less familiar with it. While it is not difficult, it is one of the main motivations for the (otherwise somewhat opaque) definition of modularity, and it seems worthwhile to state as a lemma. We broke the material on modular/normal subgroups into its own subsection, and put the proof of the Corollary in a separate subsection, as the referee recommended. We added the reference to Robinson, but kept that to Isaacs. For one thing, it helps with the seque "another easy exercise", and the result on prime-powers and quotients we did not find in Robinson. Again, all this is certainly easy and routine for group-theorists. We'd also like to be as inviting as possible for the non-group-theorists in our audience. > 8. page 5. > Please state Theorem 2.2 before the Proof of Theorem 1.2, not in the middle > of it. We kept Theorem 2.2 in the same subsection, but more clearly separated it from the proof with phrases "In order to prove Theorem 1.2, we will need ..." and "We now complete the proof of Theorem 1.2." Since the section is quite short, this should certainly cause no confusion! > 9. page 5, line 10. It would be fair to acknowledge that the large body of > work which precedes the (precise) result of King goes much beyond the > (essentially probabilistic) result of Proposition 2.3. The class of > (2,3)-generated groups is the main instance for several reasons: it includes > most finite simple groups of order divisible by 3, it coincides with the > epimorphic images (of order divisible by 6) of the unimodular group PSL2(Z), > it includes Hurwitz groups. So it deserves a little more relevance. We're not quite sure what more the referee wants here. We already mention the connection with PSL_2(Z) in this section, with some background references. Hurwitz groups are an interesting class of examples of (2,3)-generated groups, but sufficiently motivating them would take us rather far afield. We slightly expanded the paragraph mentioning PSL_2(Z), with a brief mention of a connection to Riemann surfaces. Also promoted the definition of (p,q)-generation to Definition 2.3, which puts the discussion of (2,3)-generation in a paragraph by itself. > 10. middle of page 5. PSp4(q) twice. Unless we're missing something, we believe we have this correct. There are three mentions of PSp4 -- the first says that it is exceptional, the second discusses characteristics 2 and 3, while the third discusses other characteristics. > 11. page 6, line 3. > To say ”‘For the first part of the injection”’, ”‘For the second > part of the injection” > is not a rigorous way of defining a map. This can easily be avoided, turning > the > argument into the following form: > Since [x, 1] ≤ . . . there is an injection > φ1 . . . . > (Notice that if x = y, then ...). We consider the map > φ2 : [x∨y, 1]→[ 0,m] > α → m ∧ α . > The rest remains unchanged. We made changes along these lines, and are less casual. We kept a framing comment that we construct the injection in two stages, which may help orient the reader to what is coming.