# Celebratio Mathematica

## Andrew Mattei Gleason

### Gleason’s contribution to the solution of Hilbert’s Fifth Problem

#### What is Hilbert’s Fifth Problem?

Andy Gleason is prob­ably best known for his work con­trib­ut­ing to the solu­tion of Hil­bert’s Fifth Prob­lem. We shall dis­cuss this work be­low, but first we need to know just what the “Fifth Prob­lem” is. In its ori­gin­al form it asked, roughly speak­ing, wheth­er a con­tinu­ous group ac­tion is ana­lyt­ic in suit­able co­ordin­ates. But as we shall see, the mean­ing has changed over time.

As Hil­bert stated it in his lec­ture de­livered be­fore the In­ter­na­tion­al Con­gress of Math­em­aticians in Par­is in 1900 [e2], the Fifth Prob­lem is linked to Sophus Lie’s the­ory of trans­form­a­tion groups [e1], i.e., Lie groups act­ing as groups of trans­form­a­tions on man­i­folds. The “groups” that Lie dealt with were really just neigh­bor­hoods of the iden­tity in what we now call a Lie group, and his group ac­tions were defined only loc­ally, but we will ig­nore such loc­al versus glob­al con­sid­er­a­tions in what fol­lows. However, it was cru­cial to the tech­niques that Lie used that his man­i­folds should be ana­lyt­ic and that both the group law and the func­tions de­fin­ing the ac­tion of the group on the man­i­fold should be ana­lyt­ic, that is, giv­en by con­ver­gent power series. For Lie, who ap­plied his the­ory to such things as study­ing the sym­met­ries of dif­fer­en­tial equa­tions, the ana­lyti­city as­sump­tions were nat­ur­al enough. But Hil­bert wanted to use Lie’s the­ory as part of his lo­gic­al found­a­tions of geo­metry, and for this pur­pose Hil­bert felt that ana­lyti­city was un­nat­ur­al and per­haps su­per­flu­ous. So Hil­bert asked if ana­lyti­city could be dropped in fa­vor of mere con­tinu­ity. More pre­cisely, if one only as­sumed a pri­ori that the group $$G$$ was a loc­ally Eu­c­lidean to­po­lo­gic­al group, that the man­i­fold $$M$$ was a to­po­lo­gic­al man­i­fold, and that the ac­tion of $$G$$ on $$M$$ was con­tinu­ous, could one nev­er­the­less al­ways choose loc­al co­ordin­ates in $$G$$ and $$M$$ so that both the group op­er­a­tions and the ac­tion be­came ana­lyt­ic when ex­pressed in these co­ordin­ates? We shall speak of the prob­lem in this gen­er­al­ity as the un­res­tric­ted Hil­bert Fifth Prob­lem. The re­stric­ted prob­lem is the im­port­ant spe­cial case in which $$G = M$$ and the ac­tion is left trans­la­tion. Ask­ing wheth­er we can al­ways find ana­lyt­ic co­ordin­ates in the re­stric­ted prob­lem is clearly the same as ask­ing wheth­er a loc­ally Eu­c­lidean group is ne­ces­sar­ily a Lie group.

#### Counterexamples

It turned out that there are many — and in fact many dif­fer­ent kinds of — counter­examples to the un­res­tric­ted Hil­bert Fifth Prob­lem. Per­haps the first pub­lished counter­example, due to R. H. Bing [e15], is an ac­tion of $$\mathbf{Z}^2$$ on $$\mathbf{S}^3$$ whose fixed-point set is the Al­ex­an­der Horned Sphere $$\Sigma$$. Now $$\Sigma$$ is not “tamely em­bed­ded” in $$\mathbf{S}^3$$, mean­ing that there are points where it is im­possible to choose co­ordin­ates so that loc­ally $$\Sigma$$ looks like the usu­al em­bed­ding of $$\mathbf{R}^2$$ in $$\mathbf{R}^3$$. If the ac­tion were even dif­fer­en­ti­able in some suit­able co­ordin­ates, then it is easy to see that the fixed-point set would in fact be tamely em­bed­ded. (For an even more bizarre type of counter­example, re­call that in 1960 M. Ker­vaire [e24] con­struc­ted a to­po­lo­gic­al man­i­fold that did not ad­mit any dif­fer­en­ti­able struc­ture, provid­ing what can be con­sidered a counter­example even for the case when $$G$$ is the trivi­al group.)

One could make a case that these ex­amples are “mon­sters” that could have been ruled out if Hil­bert had phrased his state­ment of the Fifth Prob­lem more care­fully. But there is a more ser­i­ous kind of counter­example that is so ele­ment­ary that it makes one won­der how much thought Hil­bert had giv­en to the Fifth Prob­lem be­fore pro­pos­ing it. Here is a par­tic­u­larly ele­ment­ary ex­ample, due to Mont­gomery and Zip­pin [e22], with $$G = \mathbf{R}$$, the ad­dit­ive group of the real num­bers, and $$M = \mathbf{C}$$, the com­plex plane. Let $$f$$ be a con­tinu­ous real-val­ued func­tion defined on the pos­it­ive real ax­is, and define the ac­tion $$\phi :\mathbf{R} \times \mathbf{C} \to \mathbf{C}$$ by $$\phi(t , r e^{i \theta} ) := r e^{i (\theta +f (r )t)}$$ . (In words, $$\phi$$ is a one-para­met­er group of homeo­morph­isms of the plane that ro­tates each circle centered at the ori­gin in­to it­self, the circle of ra­di­us $$r$$ be­ing ro­tated with an­gu­lar ve­lo­city $$f (r )$$.) If we choose $$f (r )$$ to equal 1 for $$r \leq 1$$ and 0 for $$r \geq 2$$, the ac­tion is the stand­ard one-para­met­er group of ro­ta­tions of $$\mathbf{C}$$ in­side the unit disk and is trivi­al out­side the disk of ra­di­us 2, so by the Prin­ciple of Ana­lyt­ic Con­tinu­ation, this ac­tion can­not be made ana­lyt­ic in any co­ordin­ate sys­tem. What is worse, we can choose $$f$$ to have these prop­er­ties and also be smooth (mean­ing $$C^{\infty}$$), so we see that even if we as­sume a pri­ori that the ac­tion of a Lie Group on a man­i­fold is smooth, it does not fol­low that it can be made ana­lyt­ic!

After these counter­examples to the un­res­tric­ted Hil­bert Fifth Prob­lem be­came known, a ta­cit un­der­stand­ing grew up to in­ter­pret “the Fifth Prob­lem” as re­fer­ring to the re­stric­ted ver­sion: Is every loc­ally Eu­c­lidean group a Lie group? and we shall fol­low this con­ven­tion be­low.

#### Early history of the Fifth Problem

It was fairly easy to settle the one-di­men­sion­al case. The only (para­com­pact) con­nec­ted man­i­folds of di­men­sion one are the real line, $$\mathbf{R}$$, and the circle, $$\mathbf{S}^1$$, and both of course are Lie groups. In 1909 L. E. J. Brouwer [e3] showed that a to­po­lo­gic­al group that is homeo­morph­ic to either of these is in fact iso­morph­ic to it as a to­po­lo­gic­al group. Us­ing res­ults from Brouwer’s pa­per, B. Kerékjártó [e6] settled the two-di­men­sion­al case in 1931. There seems to have been little if any pub­lished work on the Fifth Prob­lem between the pa­pers of Brouwer and Kerékjártó, but that is not too sur­pris­ing con­sid­er­ing that much of the mod­ern math­em­at­ic­al in­fra­struc­ture re­quired for a rig­or­ous dis­cus­sion of to­po­lo­gic­al groups and the Fifth Prob­lem be­came avail­able only after a 1926 pa­per by O. Schreier [e4]. The three-di­men­sion­al and four-di­men­sion­al cases of the Fifth Prob­lem were settled much later, by Mont­gomery [e12] in 1948 and by Mont­gomery and Zip­pin [e17] in 1952.

The first ma­jor break­through in the gen­er­al the­ory came in 1933, when J. von Neu­mann [e8], us­ing the re­cently dis­covered Haar [e7] meas­ure, ex­ten­ded the Peter–Weyl The­or­em [e5] to gen­er­al com­pact groups and used it to settle the Fifth Prob­lem in the af­firm­at­ive for com­pact groups. We will sketch a proof of von Neu­mann’s the­or­em be­low. Sev­er­al years later, build­ing on von Neu­mann’s work, Pontry­agin [e9] settled the abeli­an case, and Che­val­ley [e10] the solv­able case.

#### The no small subgroups (NSS) condition

The first time I en­countered the phrase “group without small sub­groups” I wondered what kind of sub­group a “small” one could pos­sibly be. Of course, what the phrase means is a to­po­lo­gic­al group without ar­bit­rar­ily small sub­groups, i.e., one hav­ing a neigh­bor­hood of the iden­tity that in­cludes no sub­group ex­cept the trivi­al group. We shall fol­low Ka­plansky [e25] and call this the NSS Con­di­tion and a group sat­is­fy­ing it an NSS group. Since NSS may seem a little con­trived, here is a brief dis­cus­sion of the “why and how” of its use in solv­ing the Fifth Prob­lem.

It turns out to be dif­fi­cult to draw use­ful con­clu­sions about a to­po­lo­gic­al group from the as­sump­tion that it is loc­ally Eu­c­lidean. So the strategy used for set­tling the Fifth Prob­lem was to look for a more group-ori­ented “bridge con­di­tion” and use it in a two-pronged at­tack: on the one hand show that a to­po­lo­gic­al group that sat­is­fies this con­di­tion is a Lie group, and on the oth­er show that a loc­ally Eu­c­lidean group sat­is­fies the con­di­tion. If these two pro­pos­i­tions can be proved, then the pos­it­ive solu­tion of the Fifth Prob­lem fol­lows — and even a little more.

As you may have guessed, NSS turned out to be ideally suited to play the role of the bridge. In ret­ro­spect this is not en­tirely sur­pris­ing. A power­ful but re­l­at­ively ele­ment­ary prop­erty of Lie groups is the ex­ist­ence of so-called ca­non­ic­al co­ordin­ates, or equi­val­ently the fact that the ex­po­nen­tial map is a dif­feo­morph­ism of a neigh­bor­hood of zero in the Lie al­gebra onto a neigh­bor­hood $$U$$ of the iden­tity in the group (see be­low). Since a line through the ori­gin in the Lie al­gebra maps to a one-para­met­er sub­group of the group, it fol­lows that such a $$U$$ con­tains no non­trivi­al sub­group and hence that Lie groups sat­is­fy NSS.

Start­ing in the late 1940s Gleason [1],1 Mont­gomery [e16], and Iwas­awa [e13] made sev­er­al sol­id ad­vances re­lated to the Fifth Prob­lem. This led in 1952 to a sat­is­fy­ing de­noue­ment to the story of the Fifth Prob­lem, with Gleason and Mont­gomery–Zip­pin car­ry­ing out the above two-pronged at­tack. First Gleason [3] proved that a loc­ally com­pact group sat­is­fy­ing NSS is a Lie group, and then im­me­di­ately af­ter­wards Mont­gomery and Zip­pin [e17] used Gleason’s res­ult to prove in­duct­ively that loc­ally Eu­c­lidean groups of any di­men­sion sat­is­fy NSS. Their two pa­pers ap­peared to­geth­er in the same is­sue of the An­nals of Math­em­at­ics, and at that point one knew that for loc­ally com­pact to­po­lo­gic­al groups: $\text{Locally Euclidean } \Longleftrightarrow \text{NSS } \Longleftrightarrow \text{Lie}.$ (Ac­tu­ally, the above is not quite the full story; Gleason as­sumed a weak form of fi­nite di­men­sion­al­ity in his ori­gin­al ar­gu­ment that NSS im­plies Lie, but shortly there­after Yamabe [e20] showed that fi­nite di­men­sion­al­ity was not needed in the proof.)

#### Cartan’s theorem

Start­ing with von Neu­mann, all proofs of cases of the Fifth Prob­lem, in­clud­ing Gleason’s, were ul­ti­mately based on the fol­low­ing clas­sic res­ult that goes back to É. Cartan. (For a mod­ern proof, see Che­val­ley ([e11], page 130).)
The­or­em (Cartan): If a loc­ally com­pact group has a con­tinu­ous, in­ject­ive ho­mo­morph­ism in­to a Lie group and, in par­tic­u­lar, if it has a faith­ful fi­nite-di­men­sion­al rep­res­ent­a­tion, then it is a Lie group.

Here is a quick sketch of how the proof of the Fifth Prob­lem for a com­pact NSS group $$G$$ fol­lows. Let $$\mathcal{H}$$ de­note the Hil­bert space $$L^2 (G)$$ of square-in­teg­rable func­tions on $$G$$ with re­spect to Haar meas­ure. Left trans­la­tion in­duces an or­tho­gon­al rep­res­ent­a­tion of $$G$$ on $$\mathcal{H}$$, the so-called reg­u­lar rep­res­ent­a­tion, and, ac­cord­ing to the Peter–Weyl The­or­em, $$\mathcal{H}$$ is the or­tho­gon­al dir­ect sum of fi­nite-di­men­sion­al sub­rep­res­ent­a­tions, $$\mathcal{H}_i$$, i.e., $$\mathcal{H} =\bigoplus^{\infty}_{i=1} \mathcal{H}_i$$. Define $$W_N := \bigoplus^N_{i=1}\mathcal{H}_i$$. We will show that for $$N$$ suf­fi­ciently large, the fi­nite-di­men­sion­al rep­res­ent­a­tion of $$G$$ on $$W_N$$ is faith­ful or, equi­val­ently, that for some $$N$$ the ker­nel $$K_N$$ of the reg­u­lar rep­res­ent­a­tion re­stric­ted to $$W_N$$ is the trivi­al group $$\{e\}$$. Since the reg­u­lar rep­res­ent­a­tion it­self is clearly faith­ful, $$K_N$$ is a de­creas­ing se­quence of com­pact sub­groups of $$G$$ whose in­ter­sec­tion is $$\{e\}$$. Thus if $$U$$ is an open neigh­bor­hood of $$e$$ that con­tains no non­trivi­al sub­group, $$K_N \backslash U$$ is a de­creas­ing se­quence of com­pact sets with empty in­ter­sec­tion and, by the defin­i­tion of com­pact­ness in terms of closed sets, some $$K_N \backslash U$$ must be empty. Hence $$K_N \subseteq U$$, and since $$K_N$$ is a sub­group of $$G$$, $$K_N = \{e\}$$.

#### Following in Gleason’s footsteps

Andy Gleason put lots of re­marks and clues in his pa­pers about his mo­tiv­a­tions and trains of thought, and it is an en­joy­able ex­er­cise to read these chro­no­lo­gic­ally and use them to guess how he de­veloped his strategy for at­tack­ing the Fifth Prob­lem.

Let’s start with a Lie group $$G$$, and let $$\mathfrak{g}$$ de­note its Lie al­gebra. There are (at least!) three equi­val­ent ways to think of an ele­ment of the vec­tor space $$\mathfrak{g}$$. First as a vec­tor $$v$$ in $$T G_e$$, the tan­gent space to $$G$$ at $$e$$; second as the left-in­vari­ant vec­tor field $$X$$ on $$G$$ ob­tained by left trans­lat­ing $$v$$ over the group; and third as the one-para­met­er sub­group $$\phi$$ of $$G$$ ob­tained as the in­teg­ral curve of $$X$$ start­ing at the iden­tity. The ex­po­nen­tial map $$\exp : \mathfrak{g} \to G$$ is defined by $$\exp(v ) = \phi(1)$$. It fol­lows im­me­di­ately from this defin­i­tion that $$\exp(0) = e$$ and that the dif­fer­en­tial of $$\exp$$ at 0 is the iden­tity map of $$T G_e$$, so by the in­verse func­tion the­or­em, $$\exp$$ maps a neigh­bor­hood of 0 in $$\mathfrak{g}$$ dif­feo­morph­ic­ally onto a neigh­bor­hood of $$e$$ in $$G$$. Such a chart for $$G$$ is called a ca­non­ic­al co­ordin­ate sys­tem (of the first kind).

Now, sup­pose we some­how “lost” the dif­fer­en­ti­able struc­ture of $$G$$ but re­tained our know­ledge of $$G$$ as a to­po­lo­gic­al group. Is there some way we could use the lat­ter know­ledge to re­cov­er the dif­fer­en­ti­able struc­ture? That is, can we re­con­struct $$\mathfrak{g}$$ and the ex­po­nen­tial map? If so, then we are clearly close to a solu­tion of the Fifth Prob­lem. Let’s listen in as Andy pon­ders this ques­tion.

“Well, if I think of $$\mathfrak{g}$$ as be­ing the one-para­met­er groups, that’s a group the­or­et­ic concept. Let’s see — is there some way I can in­vert $$\exp$$? That is, giv­en $$g$$ in $$G$$ close to $$e$$, can I find the one-para­met­er group $$\phi$$ such that $$\phi (1) = \exp(\phi) = g$$? Now I know square roots are unique near $$e$$ and in fact $$\phi(1/2)$$ is the square root of $$g$$. By in­duc­tion, I can find $$\phi(1/2^n)$$ by start­ing with $$g$$ and tak­ing the square root $$n$$ times. And once I have $$\phi(1/2^n)$$, by simply tak­ing its $$m$$-th power I can find $$\phi(m/2^n )$$ for all $$m$$. So, if I know how to take square roots near $$e$$, then I can com­pute $$\phi$$ at all the dy­ad­ic ra­tion­als $$m /2^n$$, and since they are dense in $$\mathbf{R}$$, I can ex­tend $$\phi$$ by con­tinu­ity to find it on all of $$\mathbf{R}!$$

This was the stated mo­tiv­a­tion for Gleason’s pa­per “Square roots in loc­ally Eu­c­lidean groups” [1], and in it he goes on to take the first step and show that in any NSS loc­ally Eu­c­lidean group $$G$$, there are neigh­bor­hoods $$U$$ and $$V$$ of the iden­tity such that every ele­ment in $$U$$ has a unique square root in $$V$$. Al­most im­me­di­ately after this art­icle ap­peared, in a pa­per called “On a the­or­em of Gleason”, Che­val­ley [e14] went on to com­plete the pro­gram Andy out­lined. That is, Che­val­ley used Gleason’s ex­ist­ence of unique square roots to con­struct a neigh­bor­hood $$U$$ of the iden­tity in $$G$$ and a con­tinu­ous map­ping $$(g , t ) \mapsto \phi^g (t )$$ of $$U \times \mathbf{R}$$ in­to $$G$$ such that each $$\phi^g$$ is a one-para­met­er sub­group of $$G , \phi^g (t) \in U$$ for $$|t| \leq 1$$, and $$\phi^g (1) = g$$.

In his key 1952 An­nals pa­per “Groups without small sub­groups” [3], Gleason de­cided not to fol­low up this ap­proach to the solu­tion of the Fifth Prob­lem and in­stead used a vari­ant of von Neu­mann’s meth­od. His ap­proach was based on the con­struc­tion of one-para­met­er sub­groups, but these were used as a tool to find a cer­tain fi­nite-di­men­sion­al in­vari­ant lin­ear sub­space $$Z$$ of the reg­u­lar rep­res­ent­a­tion of $$G$$ on which $$G$$ ac­ted faith­fully and ap­pealed to Cartan’s The­or­em to com­plete the proof. The con­struc­tion of $$Z$$ is a tech­nic­al tour de force, but it is too com­plic­ated to out­line here, and we refer in­stead to the ori­gin­al pa­per [2] or the re­view by Iwas­awa.

#### Andy Gleason as mentor

Look­ing back at how it happened, it seems al­most ac­ci­dent­al that I be­came Andy Gleason’s first Ph.D. stu­dent — and Dav­id Hil­bert was partly re­spons­ible.

As an un­der­gradu­ate at Har­vard I had de­veloped a very close ment­or­ing re­la­tion­ship with George Mackey, then a res­id­ent tu­tor in my dorm, Kirk­land House. We had meals to­geth­er sev­er­al times each week, and I took many of his courses. So, when I re­turned in 1953 as a gradu­ate stu­dent, it was nat­ur­al for me to ask Mackey to be my thes­is ad­visor. When he in­quired what I would like to work on for my thes­is re­search, my first sug­ges­tion turned out to be something he had thought about him­self, and he was able to con­vince me quickly that it was un­suit­ably dif­fi­cult for a thes­is top­ic. A few days later I came back and told him I would like to work on re­for­mu­lat­ing the clas­sic­al Lie the­ory of germs of Lie groups act­ing loc­ally on man­i­folds as a rig­or­ous mod­ern the­ory of full Lie groups act­ing glob­ally. Fine, he said, but then ex­plained that the loc­al ex­pert on such mat­ters was a bril­liant young former Ju­ni­or Fel­low named Andy Gleason who had just joined the Har­vard math de­part­ment. Only a year be­fore he had played a ma­jor role in solv­ing Hil­bert’s Fifth Prob­lem, which was closely re­lated to what I wanted to work on, so he would be an ideal per­son to dir­ect my re­search.

I felt a little un­happy at be­ing cast off like that by Mackey, but of course I knew per­fectly well who Gleason was and I had to ad­mit that George had a point. Andy was already fam­ous for be­ing able to think com­plic­ated prob­lems through to a solu­tion in­cred­ibly fast. “Johnny” von Neu­mann had a sim­il­ar repu­ta­tion, and since this was the year that High Noon came out, I re­call jokes about hav­ing a math­em­at­ic­al shootout — Andy vs. Johnny solv­ing math prob­lems with blaz­ing speed at the OK Cor­ral. In any case, it was with con­sid­er­able trep­id­a­tion that I went to see Andy for the first time.

Totally un­ne­ces­sary! In our ses­sions to­geth­er I nev­er felt put down. It is true that oc­ca­sion­ally when I was telling him about some pro­gress I had made since our pre­vi­ous dis­cus­sion, part­way through my ex­plan­a­tion Andy would see the crux of what I had done and say something like, “Oh! I see. Very nice! and then…,” and in a mat­ter of minutes he would re­con­struct (of­ten with im­prove­ments) what had taken me hours to fig­ure out. But it nev­er felt like he was act­ing su­per­i­or. On the con­trary, he al­ways made me feel that we were col­leagues, col­lab­or­at­ing to dis­cov­er the way for­ward. It was just that when he saw his way to a solu­tion of one prob­lem, he liked to work quickly through it and then go on to the next prob­lem. Work­ing to­geth­er with such a math­em­at­ic­al power­house put pres­sure on me to per­form at top level — and it was sure a good way to learn hu­mil­ity!

My ap­pren­tice­ship wasn’t over when my thes­is was done. I re­mem­ber that shortly after I had fin­ished, Andy said to me, “You know, some of the ideas in your thes­is are re­lated to some ideas I had a few years back. Let me tell you about them, and per­haps we can write a joint pa­per.” The ideas in that pa­per were in large part his, but on the oth­er hand, I did most of the writ­ing, and in the pro­cess of cor­rect­ing my at­tempts he taught me a lot about how to write a good journ­al art­icle.

But it was only years later that I fully ap­pre­ci­ated just how much I had taken away from those years work­ing un­der Andy. I was very for­tu­nate to have many ex­cel­lent stu­dents do their gradu­ate re­search with me over the years, and of­ten as I worked to­geth­er with them I could see my­self be­hav­ing in some way that I had learned to ad­mire from my own ex­per­i­ence work­ing to­geth­er with Andy.

Let me fin­ish with one more an­ec­dote. It con­cerns my fa­vor­ite of all Andy’s the­or­ems, his el­eg­ant clas­si­fic­a­tion of the meas­ures on the lat­tice of sub­spaces of a Hil­bert space. Andy was writ­ing up his res­ults dur­ing the 1955–56 aca­dem­ic year, as I was writ­ing up my thes­is, and he gave me a draft copy of his pa­per to read. I found the res­ult fas­cin­at­ing, and even con­trib­uted a minor im­prove­ment to the proof, as Andy was kind enough to foot­note in the pub­lished art­icle. When I ar­rived at the Uni­versity of Chica­go for my first po­s­i­tion the next year, Andy’s pa­per was not yet pub­lished, but word of it had got­ten around, and there was a lot of in­terest in hear­ing the de­tails. So when I let on that I was fa­mil­i­ar with the proof, Ka­plansky asked me to give a talk on it in his ana­lys­is sem­in­ar. I’ll nev­er for­get walk­ing in­to the room where I was to lec­ture and see­ing Ed Span­i­er, Mar­shall Stone, Saun­ders Mac Lane, An­dré Weil, Ka­plansky, and Chern all look­ing up at me. It was pretty in­tim­id­at­ing, and I was suit­ably nervous!

But the pa­per was so el­eg­ant and clear that it was an ab­so­lute breeze to lec­ture on it, so all went well, and this “in­aug­ur­al lec­ture” helped me get off to a good start in my aca­dem­ic ca­reer.

### Works

[1]A. M. Gleason: “Square roots in loc­ally Eu­c­lidean groups,” Bull. Am. Math. Soc. 55 : 4 (1949), pp. 446–​449. MR 0028841 Zbl 0041.​16002 article

[2]A. M. Gleason: “The struc­ture of loc­ally com­pact groups,” Duke Math. J. 18 : 1 (1951), pp. 85–​104. MR 0039730 Zbl 0044.​01901 article

[3]A. M. Gleason: “Groups without small sub­groups,” Ann. Math. (2) 56 : 2 (September 1952), pp. 193–​212. MR 0049203 Zbl 0049.​30105 article