## At the third stroke

**Further update 2022-12-06:** after I wrote the first version of this post, I discovered an extra reference that I should have cited in the bibliography, and which would have made my life easier had I known about it before submitting! See the body of the post for further details.

An update: the paper “An explicit minorant for the amenability constant of the Fourier algebra”, which has been mentioned in previous posts on 2020-12-29 and 2021-12-29, has recently been accepted for publication in International Mathematical Resarch Notices IMRN (the redundancy in the name is not a typo).

The “author accepted manuscript”, as we must now call these things in the UK’s Procrustean REF-linked open access framework, can be found on the arXiv as 2012.14413. Since the abstract has been tweaked slightly compared to the submitted ~~previous~~ version, here is the updated abstract:

We show that if a locally compact group G is non-abelian then the amenability constant of its Fourier algebra is ≥ 3/2, extending a result of Johnson (JLMS, 1994) who proved that this holds for finite non-abelian groups. Our lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by Runde (PAMS, 2006).

To do this we study a minorant for the amenability constant, related to the anti-diagonal in G× G, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when G is a countable virtually abelian group, in terms of the Plancherel measure for G. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value, and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

The accepted version benefited significantly from a detailed referee’s report, which suggested some technical improvements in the discussion of measurability issues, and also gave a sketch of the following result: if A and B are C^{*}-algebras, and D denotes the completion of A ⊗ B in some C^{*}-tensor norm, then the canonical map is a homeomorphism onto its range. (There are known examples, with both A and B beng C^{*}-algebras of discrete groups, where the range of this map is not closed in the codomain.) The application relevant to this paper is as follows: applying this result when A and B are the full group C^{*}-algebras of Type I groups G and H, in conjuction with some standard results, one deduces that the unitary dual of G × H can be naturally identified (as a topological space) with the product of the respective duals. (Somewhat irritatingly, this conclusion is not stated anywhere in Dixmier’s C^{*}-book, even though I suspect it must have been known to people working on group C^{*}-algebras at the time the book was written.)

**Update 2022-12-06:** it turns out that the C^{*}-algebraic result referred to above can be found in one of the appendices to Raeburn and Williams’s book on Morita Equivalence and Continuous-Trace C^{*}-Algebras, which is not a book I own and not a source I would have thought to turn to. The arguments they use are essentially the same as the one I came up with in the appendix of my paper based on the referee’s sketch. However, since they develop various things from scratch to keep their account more self-contained, I decided to keep the appendix to my paper (which outsources a lot of standard details to Dixmier’s book) and merely add a note to acknowledge the Raeburn-Williams book.

In an earlier version of the paper (and in the 2021-12-29 blogpost) I mentioned the following question: *if G is a finite group whose amenability constant is less than or equal to 2, do all its irreps have degree at most 2?* It turned out, in between the submission and the acceptance, that the answer is yes, but I decided to leave that for a future paper (perhaps for future blog posts).

## Divisibility testing

A minor bit of throwaway blogging, just as displacement activity.

For several years I have been joking with colleagues that given any 3 digit number I start trying to find its prime factorization, in case it can be used as the raw input for an exercise/exam question on the Number Theory course that I have been teaching at Lancaster.

Well, I was recently staying in a hotel room numbered 647, and I did indeed find myself thinking “hmm, I wonder if that’s prime, it smells like it should be prime”. It then struck me that this particular number is a useful illustration of some tricks that can be useful to speed up “testing for divsibility by primes less than 30”, which I am not sure I ever actually explained in class. None of this is original, and I am sure has been re-invented by generations of students who either didn’t have hand-held calculators or just enjoyed being able to play around with mental arithmetic. But it is an illustration of how “long division with remainder / the Euclidean algorithm” is often not the quickest way to do things for small numbers.

So: as usual we start by noting that 647 is odd (not divisible by 2) and its digit sum is 2 mod 3 (hence 647 itself is 2 mod 3). It is an odd number which doesn’t end in 5, so it is not divisible by 5.

What about divisibility by 7? Rather than doing long division, note that 647 is divisible by 7 if and only if 647-7=640 is. But since 7 is coprime to 10, it divides 640 if and only if it divides 64, and we rule this out by remembering that 64 is a power of 2.

Divisibility by 11 is the standard “alternating digit sum” test: 6-4+7 is not 0 mod 11, so 647 is not divisible by 11.

What about divisibility by 13? Again, rather than doing long division we try adding/subtracting multiples of 13 to 647 until we get a multiple of 10, then arguing as we did in the “divisibility by 7” case. Thus: 647+13 = 660 so 13 divides 647 if and only if it divides 66, and we know this isn’t the case since 66 =2 . 3. 11. Similarly for 17: since 647-17=630 and 17 doesn’t divide 63= 3^2 . 7, we see that 647 is not divisible by 17.

Recalling that 25^2 < 647 < 26^2 we see that it only remains to check divisibility by 19 or 23. Since 57 is a multiple of 19 and 647 – 57 = 590, 59-19 – 40, we see that 19 does not divide 647. Finally, since 647+23 =670 and 67 + 23 = 90, we see that 23 does not divide 647.

We conclude that 647 is indeed prime.

Having gone through all this, it strikes me that I never looked into the algorithmic (in)efficiency of the following approach to testing if a given N is divisibly by a given odd prime p. Namely, inductively define and stop when we reach .

For N=647 and p = 7 this would have yielded the sequence . For N=647 and p = 19 the corresponding sequence is

.

## They gave Boxer a medal, didn’t they?

And Yossarian, too, if memory serves correctly.

If a power was to lift him up,

Make him rich, would he admit it was luck?

Or say he’d earned it, claim a state of grace,

Just like the rich in this hateful place

## as is once, will always be

together alone, shallow and deep,

holding our breath, paying death no heed;

I am still your friend when you are in need —

as is once, will always be,

earth and sky, moon and sea

The French have a phrase for it. The bastards have a phrase for everything and they are always right. To say goodbye is to die a little.

— from The Long Goodbye by Raymond Chandler —

## Uh muh nuh muh (do doo do do do)

Continuing a theme from earlier posts, of finding myself too drained/busy to blog about my recent work but also feeling that in this day and age one must develop the habit of announcing/recording one’s activity…

Bence Horváth, Niels Laustsen and I have recently had the following paper accepted by the Transactions of the American Mathematical Society:

## Approximately multiplicative maps between algebras of bounded operators on Banach spaces

Coauthors: B. Horváth, N. J. Laustsen

We show that for any separable reflexive Banach space

Xand a large class of Banach spacesE, including those with a subsymmetric shrinking basis but also all spacesLfor 1 ≤_{p}p≤ ∞, every bounded linear map ℬ(E) → ℬ(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism ℬ(E) → ℬ(X). That is, the pair (ℬ(E), ℬ(X)) has the AMNM property in the sense of Johnson (J. London Math. Soc. 1988). Previously this was only known forE=X= lwith 1 <_{p}p< ∞; even for those cases, we improve on the previous methods and obtain better constants in various estimates. A crucial role in our approach is played by a new result, motivated by cohomological techniques, which establishes AMNM properties relative to an amenable subalgebra; this generalizes a theorem of Johnson (op cit.).

The main headline results mentioned in the abstract were obtained by Bence during his PhD studies at Lancaster with Niels and myself. The paper itself, however, takes a slightly different perspective from the original proofs that Bence worked out, in that a large portion of the paper is devoted to a framework of *approximate splitting for an approximate cochain complex*, which we use to prove an AMNM result relative to an amenable subalgebra. This is an extrapolation from a sketch given by Barry Johnson in his 1988 paper mentioned above, but there he chose to do the main calculation by hand rather than setting up approximate cochain complexes explicitly. My coauthors and I felt that having this machinery written out explicitly would be useful for future researchers.

Strictly speaking, Johnson’s sketch and result apply to AMNM problems where the domain algebra is amenable; our “relative version” is influenced by ideas concerning normalization of (Hochschild) cocycles with respect to an amenable subalgebra, a theme that has been well known to specialists and used in work of many authors (Johnson-Kadison-Ringrose; Christensen-Sinclair-Smith; Lykova; Grønbæk; …), and which I learned long ago from Sinclair-Smith’s book on Hochschild Cohomology of von Neumann Algebras.

The paper had an unusually long gestation time, for which I probably bear most of the blame. Fortunately, my coauthors have been forgiving and the refereeing time was relatively short.

Finally, credit (or blame) for the title of the blog post must go to Mahya Ghandehari and Michael White who independently pointed out to me some years ago that calling something AMNM elicits a Pavlovian response.

## Of the golden future time

(All excerpts provided thanks to Project Gutenberg.)

But by and large the animals enjoyed these celebrations. They found it comforting to be reminded that, after all, they were truly their own masters and that the work they did was for their own benefit. So that, what with the songs, the processions, Squealer’s lists of figures, the thunder of the gun, the crowing of the cockerel, and the fluttering of the flag, they were able to forget that their bellies were empty, at least part of the time.

It was only his appearance that was a little altered; his hide was less shiny than it had used to be, and his great haunches seemed to have shrunken. The others said, “Boxer will pick up when the spring grass comes on”; but the spring came and Boxer grew no fatter. Sometimes on the slope leading to the top of the quarry, when he braced his muscles against the weight of some vast boulder, it seemed that nothing kept him on his feet except the will to continue. At such times his lips were seen to form the words, “I will work harder”; he had no voice left. Once again Clover and Benjamin warned him to take care of his health, but Boxer paid no attention. His twelfth birthday was approaching. He did not care what happened so long as a good store of stone was accumulated before he went on pension.

Napoleon himself appeared at the meeting on the following Sunday morning and pronounced a short oration in Boxer’s honour. It had not been possible, he said, to bring back their lamented comrade’s remains for interment on the farm, but he had ordered a large wreath to be made from the laurels in the farmhouse garden and sent down to be placed on Boxer’s grave. And in a few days’ time the pigs intended to hold a memorial banquet in Boxer’s honour. Napoleon ended his speech with a reminder of Boxer’s two favourite maxims, “I will work harder” and “Comrade Napoleon is always right”–maxims, he said, which every animal would do well to adopt as his own.

The farm was more prosperous now, and better organised: it had even been enlarged by two fields which had been bought from Mr. Pilkington. The windmill had been successfully completed at last, and the farm possessed a threshing machine and a hay elevator of its own, and various new buildings had been added to it. Whymper had bought himself a dogcart. The windmill, however, had not after all been used for generating electrical power. It was used for milling corn, and brought in a handsome money profit. The animals were hard at work building yet another windmill; when that one was finished, so it was said, the dynamos would be installed. But the luxuries of which Snowball had once taught the animals to dream, the stalls with electric light and hot and cold water, and the three-day week, were no longer talked about. Napoleon had denounced such ideas as contrary to the spirit of Animalism. The truest happiness, he said, lay in working hard and living frugally.

Somehow it seemed as though the farm had grown richer without making the animals themselves any richer — except, of course, for the pigs and the dogs. Perhaps this was partly because there were so many pigs and so many dogs. It was not that these creatures did not work, after their fashion. There was, as Squealer was never tired of explaining, endless work in the supervision and organisation of the farm.

An uproar of voices was coming from the farmhouse. They rushed back and looked through the window again. Yes, a violent quarrel was in progress. There were shoutings, bangings on the table, sharp suspicious glances, furious denials. The source of the trouble appeared to be that Napoleon and Mr. Pilkington had each played an ace of spades simultaneously.

## Mind the gap – updated

Five submissions in 2020, five publications in 2021. Not sure what can be inferred from a somewhat stochastic process, but at least I no longer hear the wolves at the door.

26. Y. Choi. A gap theorem for the ZL-amenability constant of a finite group. Int. J. Group Th. 5 (2016) no. 4, 27–46.

27. Y. Choi. Realization of compact spaces as cb-Helson sets. Ann. Funct. Anal. 7 (2016), no. 1, 158–169.

28. Y. Choi. Triviality of the generalized Lau product associated to a Banach algebra homomorphism. Bull. Austral. Math. Soc. 94 (2016), no. 2, 286–289.

29. Y. Choi, M. Ghandehari, H. L. Pham. Stability of characters and filters for weighted semilattices. Semigroup Forum 102 (2021), no. 1, 86–103.

30. Y. Choi, M. Ghandehari, H. L. Pham. Unavoidable subprojections in union-closed set systems of infinite breadth. European J. Combin. **94** (2021), article 103311 (17 pages).

31. Y. Choi. Constructing alternating 2-cocycles on Fourier algebras. Adv. Math. **385** (2021) article 107747 (28 pages).

32. Y. Choi, M. Ghandehari. Dual convolution for the affine group of the real line. Complex Anal. Oper. Theory **15** (2021), no. 4, article 76 (32 pages).

33. Y. Choi. A short proof that B(L_{1}) is not amenable. Proc. Roy. Soc. Edinburgh Sect. A **151** (2021), no. 6, 1758–1767.

## De Chelonian Mobile

This is the one year anniversary of this blogpost, announcing an arXiv posting of an article which had not yet been submitted. The arXiv posting was born from a (justified) fear that if I didn’t get some work I had done in 2020 written up and posted to the arXiv, it would just sit on my TO DO list for 2021 and beyond…

In that post, I grumbled that:

In more easy-going times, professionally speaking, I would have preferred to sit on this and use the summer of 2021 to push the techniques in the paper to their limits. But, well, these days one has to keep the wolf from the door.

As it turned out, during March and April 2021, some new ideas emerged as a result of the teaching I had been doing, alluded to here. (This was not so much a case of finding inspiration as seeking escape/relief!) I’ll say more about the new material below.

In addition, I realised during the summer of 2021 that some technical issues which I had overlooked or brushed under the carpet, concerning the unitary dual of the Cartesian square of certain groups, needed to be treated more carefully: this is related to the phenomenon/issue, discussed in various places on Kevin Buzzard’s blog, that as professional mathematicians we get increasingly comfortable – but also too cavalier – about assuming that two things which “surely must be isomorphic in a canonical way” are actually equal.

So, after a couple of months dealing with other demands, and several spells of tearing my hair out over various “proofs by reference to earlier results, or results in other books”, I finally managed to put together a proper revised version of the preprint, posted to arXiv just before Christmas.

## An explicit minorant for the amenability constant of the Fourier algebra

For a locally compact group G, let AM(A(G)) denote the amenability constant of the Fourier algebra of G. We show that AM(A(G)) ≥ 3/2 for every non-abelian G, extending a result of Johnson (JLMS, 1994) who obtained this for finite non-abelian groups. This lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by Runde (PAMS, 2006).

To do this we study a minorant for AM(G), related to the anti-diagonal in G×G, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when G is a countable virtually abelian group, in terms of the Plancherel measure for G. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value, and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

## Some remarks on the new material

Although I took the opportunity to do a lot of minor rewrites to the exposition, the most significant change in v2 compared to v1, mathematically speaking, is that the first version merely stopped at showing that AD(Γ) ≥ 3/2 for every non-abelian Γ, while the second version includes a new section that characterizes those Γ for which equality occurs: they are exactly those groups in which the centre has index 4. Such groups also have the property that the derived subgroup has order 2, and hence in various ways they can be thought of as the non-abelian groups that are closest to being abelian.

Originally I worked this out for the finite case (see this blogpost for a self-contained exposition), since this is the setting where we know that the minorant AD actually coincides with the amenability constant of the Fourier algebra. However, since I am hopeful that the two invariants agree for infinite groups as well, it seemed worthwhile to work out as much as possible for AD, for future use/interest.

The proof of the general case is technically more involved than the finite case, since we can no longer use arguments based on counting conjugacy classes. One theme, which didn’t quite survive into the final write-up, but did play an important heuristic role, is that certain statements about the centre and particular centralizers are *countably determined*, i.e. if they are true for all countable subgroups of Γ then they are true for Γ itself. This was important because the main formula in the paper for AD(Γ) was only proved under a countability assumption; it seems likely that the assumption could be removed, but one would have to reinvent or recheck a lot of material on direct integrals of group representations in order to ensure that the relevant technical machinery still remains valid.

The new version of the paper also includes a new section with some questions raised by this work (the most obvious, and the most important, being whether AD is always equal to the amenability constant of the Fourier algebra). One question which might be worth mentioning here for those interested in finite group theory: if G is a finite group and AD(G) ≤ 2, does it follow that maxdeg(G) ≤2? (This condition on degrees is clearly sufficient to ensure AD(G) ≤ 2; and dihedral groups show that we can get as close to 2 from below as we wish.) A related question, which I didn’t include explicitly in the paper: *what is the infimum of AD(G) over all finite groups G which have maxdeg ≥ 3?*

## it’s all we’re skilled in

with all the will in the world

diving for dear life

when we could be diving for pearls

## Can’t start a fire without a spark

Alternative title: “an analyst does some finite group theory, latest in an irregular and infrequent series”. Or: when life gives you lemons, get your engineers to invent a combustible lemon …

Shortly after posting the preprint mentioned in this blogpost to the arXiv, at the tail end of 2020, I started to find minor errors or omissions in the exposition; none of them affect the validity of the main results, but it did mean that some major revisions were needed. Unfortunately, the demands of teaching and assessment and quality assurance (the joys of UKHE) left me with little time and even less energy to focus on doing so.

Instead, since I was teaching a first course on groups (and rings), I found myself playing around with character theory of finite groups to keep myself entertained, and it was during one of these spells when I realized how a result in that paper could be sharpened for finite groups. Subsequently, I worked out a way to obtain the sharper result for all virtually abelian groups, but the proof becomes more technical and has more moving parts, because one does not have access to the same counting arguments that are possible for finite groups.

The proof for the general case is being written up, and will be added to the revised version of the arXiv preprint. But it seemed worth writing up the simpler argument for the finite case, for two reasons: increased forgetfulness as I get older; and increased sentimental attachment as I get older. So here it is. Probably it should have been split over several posts, but then that would have greatly decreased the chances of ever finishing it.

For a finite group G, let Irr(G) be the set of irreducible characters of G (throughout, we are working over the complex numbers, so no modular representation theory to see here). If then we write for the degree of φ, or equivalently, the dimension of any (irreducible) representation of which φ is the trace. A fundamental result in the classical character theory of finite groups is that .

Consider the following quantity, which emerged in work of Johnson (J. London Math. Soc. 1994) on the (non-)amenability of Fourier algebras of compact groups, but which in the case of finite groups can be defined directly:

**Remark 1.** The notation AD stands for “antidiagonal”, and does not appear in Johnson’s original paper. The reason for this notation/terminology comes from the 2020 work mentioned above, where AD(G) is defined in a more abstract way — but to keep this blog post more focused, I will not go into this here.

It is my belief that this numerical invariant of G deserves further study. It is fairly obvious from the definition above that . What is much less obvious, to me at least, is that whenever H is a subgroup of G, we have AD(H) ≤ AD(G). Indeed, I do not know of a direct proof using the definition above, given that the representation theory of a subgroup of G can be radically different from that of G itself; the only proof I know goes via the “abstract definition” that is alluded to above.

So what can we say about AD(G)? Let us start with some results that can be found in Johnson’s 1994 paper. First of all, since for all φ,

and this inequality is strict unless for all . That is:

**Proposition 2.**

For any finite group , we have . Equality holds if and only if is abelian.

This raises a natural question: if G is a non-abelian finite group, how small can be?

On the face of it, is conceivable that one could find non-abelian finite groups such that . But inspecting the argument used to prove Proposition 2, one sees that for to be close to 1, there must be a high proportion of 1-dimensional characters among the elments of Irr(G). Now every 1-dimensional character factors through the abelianization ; more precisely, the 1-dimensional characters of G correspond to the group homomorphisms , and hence by Fourier analysis for finite *abelian* groups, there are exactly of these characters. Since is (isomorphic to) the quotient of G by its commutator subgroup [G.G], and since [G,G] has size at least 2 when G is non-abelian, we see that

and from here, it is a short step to the following result.

**Theorem 3.** If is a non-abelian finite group, then .

Before explaining the proof of the theorem, we note the following corollary.

**Corollary 4.** If are finite non-abelian groups, then .

In particular, by taking powers of some fixed class-2 nilpotent group, we see that there are class-2 nilpotent groups with arbitrarily large AD-constant.

*The proof of Theorem 3.* Let denote the set of irreducible characters of G which have degree *n*, so that

Thus

and rearranging gives . But since is non-abelian, we know from the remarks before the theorem that , and the result follows.

So far, all of this is in Johnson’s original paper. However, he stops short of characterizing those non-abelian G for which the extremal value AD(G)=3/2 is attained. Inspecting the proof of Theorem 3, one sees that in order for AD(G) to equal 3/2, the following conditions are both necessary and sufficient:

- Equality must hold in (*);
- .

In turn, these are equivalent to the following pair of necessary and sufficient conditions:

- is empty for all , i.e. every has degree 1 or 2;

It turns out that these conditions can be replaced with one that is purely group-theoretic, i.e. one that does not make any reference to characters.

**Theorem 5.** Let G be a finite group. TFAE:

- AD(G) = 3/2;
- G/Z(G) is isomorphic to the Klein-four group.

The last condition may be phrased as: G is a non-trivial central extension of . It should therefore be possible to classify all possible G using the methods of group cohomology, but I haven’t made serious efforts to look up the necessary details.

**Remark 6.** Instinctively, one feels that should be easier than , because 2) specifies some structural property of G and 1) is a statement about some numerical invariant of G defined in terms of its representation theory. However, during my original investigations in March/April 2021, it was the implication which came first, because as we will see it follows quite easily if one takes for granted certain basic facts about the character table of a finite group. The converse implication is conceptually easy — since Z(G) has small index in G, the commutators in G must have a restricted form — but somehow writing out the proof involves more nitpicking over cases than I expected.

#### Proving the implication

We start from the “second set of conditions” stated above, and use them to show that . If we can show this, then G/Z(G) is a group of order 4; and up to isomorphism the only groups of order 4 are and the Klein-four group. But now we have a general group-theoretic fact: if H is any group (not necessarily finite), N is a subgroup of Z(H), and H/N is cyclic, then H is abelian. (Students who I was teaching in the first half of 2021 may recognize this from the mock exam paper!) In particular, in our setting cannot be cyclic, so it has to be isomorphic to the Klein-four group.

Let us now show that . The idea here is quite natural if one has ever spent time playing with character tables; note that by the given assumptions on G, all irreducible characters have degree 1 or 2, and we have a great deal of control on the number of degree 1 characters. To be precise, using the notation above, we know that and we also know that . Since | [G,G] | =2, the order of is half the order of G, and by Fourier/Pontryagin duality this means . Substituting this into the previous equations we obtain

On the other hand: let us try to count conjugacy classes in G. In any group H, there is an injection from each conjugacy class into the set of commutators of H, so in particular the size of every conjugacy class is bounded above by the order of the commutator subgroup. Since we are assuming that |[G,G]|=2, it follows that each conjugacy class of G is either a singleton (i.e. an element of the centre) or has size exactly 2. Writing for the number of conjugacy classes of size 2, we have and

**But the character table is a square!** That is, (one of the most notorious “unnatural bijections” in algebra). We therefore have

and rearranging gives as required.

#### Proving the implication

It suffices to show that 2 implies both of the “second set of conditions“. Pick a non-central element , and let be the subgroup of G generated by and Z(G). Then H is contained in the centralizer , which is a proper subgroup of G since is non-central, and so . On the other hand, Z(G) is a proper subgroup of H since , and so . Hence

Since equality must hold throughout, we see that . Now is abelian, so all its irreducible characters have degree 1. If , then by considering and invoking Frobenius reciprocity, we see that φ must occur as a summand of some induced character where . Since H is abelian , and so . This establishes the first of the two conditions.

It remains to show that . What follows is not the original proof that I came up with — see Remark 7 below — but it preserves more symmetry, at the expense of being less direct.

Let be the quotient map. The Klein-four group has the following properties:

- each non-identity element has order 2 (which by a favourite exercise of those teaching group theory, implies the group is commutative);
- any two distinct non-identity elements generate the whole group.

Now partition into four distinct cosets of , labelled as . For convenience, let denote the equivalence relation “*x* and *y* belong to the same coset of Z(G)” (equivalently, ). Then the properties of the Klein-four group listed above imply:

- ;

The equivalence relation respects multiplication (since *q* is a homomorphism) and so we also have , etc.

**The key point:** a little thought (or exercise) shows that if and then . (Here it is important that we are quotienting by Z(G) and not by some arbitrary normal subgroup.)

Since and , we have

;

and since , we also have

Everything said thus far remains invariant under a permutation of the symbols , since this just corresponds to a relabelling of the non-trivial cosets of Z(G). Therefore, starting from the identities , we obtain

Thus all six expresions are equal to the same element of G, which we denote by . Since every element of G is equivalent to one of , we have shown that for all .

Note that . For since , there exist with . But is equivalent to exactly one element of the set and is equivalent to another element of , so .

Moreover,

which shows that is an involution. Thus and we have shown that as required.

**Remark 7.** The proof just given avoided “breaking symmetry”, in the sense that we did not privilege any of the non-trivial cosets of Z(G) over any of the others; equivalently, when considering G/Z(G) we did not pick two specific generators. However, it did rely on some ad hoc trickery using the equivalence relation to show that all non-trivial commutators take the same value. In this context, it should be admitted that the argument above is not the one I first came up with when first proving in Theorem 5. The original argument went as follows: fix two noncommuting elements and in G, and note that their images in G/Z(G) commute and generate the whole group, so that G can be written as a disjoint union with . Writing one has ; squaring both sides, and using centrality of and , we obtain

Thus is an involution. I then did a case by case analysis of the various commutators , etc. and used manipulations similar to those above to show that all non-trivial commutators were equal to . Compared with the proof above, this approach feels slightly more “hacky”, but it does seem to suggest more naturally why one might hope to reduced all the commutators to (expressions involving) .