I had been looking lately at Sylow subgroups of some specific groups and it got me to wondering about why Sylow subgroups exist. I'm very familiar with the proof of the theorems (something that everyone learns at the beginning of their abstract algebra course) — incidentally my favorite proof is the one by Wielandt — but the statement of the three Sylow theorems still seems somewhat miraculous. What got Sylow to imagine that they were true (especially the first — the existence of a sylow subgroup)? Even the simpler case of Cauchy's theorem about the existence of an element of order $p$ in a finite subgroup whose order is a multiple of $p$ although easy to prove (with the right trick) also seems a bit amazing. I believe that sometimes the hardest part of a proving a theorem is believing that it might be true. So what can buttress the belief for the existence of Sylow subgroups?
[Math] Sylow Subgroups
gr.group-theorysoft-question
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The answer is no in general. I.e, there are finite non-isomorphic groups G and H such that there exists a bijection between their elements which also induces a bijection between their subgroups.
For this, I used two non-isomorphic groups which not only have the same subgroup lattice (which certainly is necessary), but also have the same conjugacy classes. There are two such groups of size 605, both a semidirect product $(C_{11}\times C_{11}) \rtimes C_5$ (see this site for details on the construction). In the small group library of GAP, these are the groups with id [ 605, 5 ] and. [ 605, 6 ]. These are provably non-isomorphic (you can construct the groups as described in the reference I gave, and then use GAPs IdSmallGroup command to verify that the groups described there are the same as the ones I am working with here). With a short computer program, one can now construct a suitable bijection.
First, let us take the two groups:
gap> G:=SmallGroup(605, 5);
<pc group of size 605 with 3 generators>
gap> H:=SmallGroup(605, 6);
<pc group of size 605 with 3 generators>
The elements of these groups are of order 1, 5 or 11, and there are 1, 484 and 120 of each. We will sort them in a "nice" way (that is, we try to match each subgroup of order 5 to another one, element by element) and obtain a bijection from this. First, a helper function to give us all elements in "nice" order:
ElementsInNiceOrder := function (K)
local elts, cc;
elts := [ One(K) ];
cc := ConjugacyClassSubgroups(K, Group(K.1));
Append(elts, Concatenation(List(cc, g -> Filtered(g,h->Order(h)=5))));
Append(elts, Filtered(Group(K.2, K.3), g -> Order(g)=11));
return elts;
end;;
Now we can take the elements in the nice order and define the bijection $f$:
gap> Gelts := ElementsInNiceOrder(G);;
gap> Helts := ElementsInNiceOrder(H);;
gap> f := g -> Helts[Position(Gelts, g)];;
Finally, we compute the sets of all subgroups of $G$ resp. $H$, and verify that $f$ induces a bijection between them:
gap> Gsubs := Union(ConjugacyClassesSubgroups(G));;
gap> Hsubs := Union(ConjugacyClassesSubgroups(H));;
gap> Set(Gsubs, g -> Group(List(g, f))) = Hsubs;
true
Thus we have established the claim with help of a computer algebra system. From this, one could now obtain a pen & paper proof for the claim, if one desires so. I have not done this in full detail, but here are some hints.
Say $G$ is generated by three generators $g_1,g_2,g_3$, where $g_1$ generates the $C_5$ factor and $g_2,g_3$ generate the characteristic subgroup $C_{11}\times C_{11}$. We choose a similar generating set $h_1,h_2,h_3$ for $H$. We now define $f$ in two steps: First, for $0\leq n,m <11$ it shall map $g_2^n g_3^m$ to $h_2^n h_3^m$.
This covers all elements of order 1 or 11, so in step two we specify how to map the remaining elements, which all have order 5. These are split into four conjugacy classes: $g_1^G$, $(g_1^2)^G$, $(g_1^3)^G$ and $(g_1^4)^G$. We fix any bijection between $g_1^G$ and $h_1^H$ and extend that to a bijection on all elements of order 5 by the rule $f((g_1^g)^n)=f(g_1^g)^n$. With some effort, one can now verify that this is a well-defined bijection between $G$ and $H$ with the desired properties. You will need to determine the subgroup lattice in each case; linear algebra helps a bit, as well as the fact that all subgroups have order 1, 5, 11, 55, 121 (unique) or 605. I'll leave the details to the reader, as I myself am happy enough with the computer result.
UPDATE: as pointed out in another answer below by @dvitek (explained by @Ian Agol in comments), there is actually a much simpler example, which I somehow overlooked when I did my computer search. Credit to them, but just in case people want to reproduce their example with GAP, here is an input session doing just that:
gap> G:=SmallGroup(16,5);; StructureDescription(G);
"C8 x C2"
gap> H:=SmallGroup(16,6);; StructureDescription(H);
"C8 : C2"
gap> Gelts := ListX([1..8],[1,2],{i,j}->G.1^i*G.2^j);;
gap> Helts := ListX([1..8],[1,2],{i,j}->H.1^i*H.2^j);;
gap> f := g -> Helts[Position(Gelts, g)];;
gap> Gsubs := Union(ConjugacyClassesSubgroups(G));;
gap> Hsubs := Union(ConjugacyClassesSubgroups(H));;
gap> Set(Gsubs, g -> Group(List(g, f))) = Hsubs;
true
Baer–Suzuki: The subgroup generated by {x,x^g} is a p-group for all g in G if and only if x is contained in the p-core of G.
Baer's proof emphasized commutators, rather than subnormality. In some sense these are the same thing, but perhaps it will feel different enough for you. Baer's presentation is given in textbook form in Huppert's Endliche Gruppen as III.6.15, page 298. See also IX.7.8 in Huppert–Blackburn, Finite Groups, Vol 2, p. 500. Baer's original paper is:
- Baer, Reinhold. "Engelsche elemente Noetherscher Gruppen." Math. Ann. 133 (1957), 256–270. MR86815 DOI:10.1007/BF02547953
Suzuki's proof is given in Gorenstein's Finite Groups, 3.8.2, p. 105. It also avoids subnormality, rather using ideas about fusion of p-elements, and is probably how Bender thought of it.
Subnormality is a pretty critical idea, and many of Bender's insights use subnormality, so I would not suggest avoiding subnormality. Other characterizations of the Fitting subgroup in terms of subnormality are given in Huppert's textbooks. In particular, the Fitting subgroup as the elements that centralize chief factors is a very important viewpoint. It generalizes to F-subnormality in the finite soluble world, and Bender's p*-nilpotency in the finite insoluble world. Kegel and Carter have a number of nice papers that explore subnormality in ways that have heavily influenced both the soluble and the insoluble worlds.
Robinson's group theory textbook (and Lennox–Stonehewer MR902857) have a good description of subnormality in the infinite case.
Wielandt's collected works contains several good textbook style presentations of subnormality that are not properly contained in any other works that I have found. They avoid assuming finiteness, and tend to have very interesting relationships between perfect subgroups and subnormality, that complement Bender's work.
One very nice thing about Isaacs's FGT is how it exposes you to important techniques. It does not try to "optimize" the presentation either by using the bare minimum of tools or by using the most general tool here-to-fore created. It just uses some nice results in a realistic way that more people should know. Suzuki's textbooks also have this nice property, though they are not as easy to quote from.
Best Answer
Victor, you should check out Sylow's paper. It's in Math. Annalen 5 (1872), 584--594. I am looking at it as I write this. He states Cauchy's theorem in the first sentence and then says "This important theorem is contained in another more general theorem: if the order is divisible by a prime power then the group contains a subgroup of that size." (In particular, notice Sylow's literal first theorem is more general than the traditional formulation.) Thus he was perhaps in part inspired by knowledge of Cauchy's theorem.
Sylow also includes in his paper a theorem of Mathieu on transitive groups acting on sets of prime-power order (see p. 590), which is given a new proof by the work in this paper. Theorems like Mathieu's may have led him to investigate subgroups of prime-power order in a general finite group (of substitutions).