The purpose of this post is to collect some proof of known results on Sylow subgroups of symmetric groups that are scattered across the literature, and in one case, wrongly stated in the standard reference. Throughout permutations act on the right. Let denote the symmetric group on .
Let be the rooted -ary tree with levels numbered from at the root to at the leaves. Let denote the set of vertices at level . We label the vertices in by , so that is a child of if and only if mod . Thus the children of are
and is a descendant of if and only if mod . For example, the children of the root vertex are labelled and correspond to congruence classes modulo , the children of are labelled (in base ) by , and correspond to the congruence classes modulo of numbers congruent to modulo , and so on. All this may be seen from the tree for shown below.
Let be the group of automorphisms of . Any automorphism is determined by its action on the leaves of , so, when it is useful to do so, we regard as a subgroup of the symmetric group .
Let . For each vertex , let fix and all vertices of that are not a descendant of , and permute the descendants of (of any generation) by adding modulo . More formally, the vertex where and is sent to . For example, if then and
(We rely on the length of the -ary form of the vertex to indicate the intended level.) Let be the subgroup of generated by all the for . Each subgroup is normalized by the for , so is a -subgroup of . We have where
It is well known that the highest power of dividing is , so is a Sylow -subgroup of .
We now make the connection with the other standard construction of . Let be the group homomorphism defined by restricting to . Thus describes the action of on the vertices at level , or, equivalently, on the congruence classes of numbers in modulo .
Lemma 1. is permutation isomorphic to the iterated wreath product with factors, in its intransitive action.
Proof. By induction is permutation isomorphic to the imprimitive wreath product with factors. Therefore is the base group in a semidirect product permutation isomorphic to in its imprimitive action.
Centre and normalizer
The centre of and its normalizer in can be described by slightly more involved inductive arguments. The following elements of will be useful: for , let . For example, if and then
Observe that for , and that .
Lemma 2. .
Proof. If then generates . Let and let . By induction . Hence . Let where and . Since is abelian, we see that commutes with all permutations in . If then there exists such that and so
a contradiction. Therefore and where for each . A similar argument considering the actions of the generators for now shows that is independent of , and so .
The normalizer turns out to be a split extension of . Let be a primitive root modulo . Let . Let fix and every vertex that is not a descendant of and permute the descendants of (of any generation) by sending where and to . Thus fixes the child of , and permutes the remaining children by a -cycle, acting compatibly on their descendants in turn. Let . For example, if and then , and
The permutation is shown below.
Proposition 3. .
Proof. Let . By a similar argument to Lemma 1, we have where . By composing with a suitable power of , we may assume that . Let denote the permutation induced by on . Since is a characteristic subgroup of , it is normalized by . Therefore normalizes the -cycle forming part of the disjoint cycle form of . Therefore acts on as some power of . Since normalizes , it follows that acts compatibly on all the other -cycles in the disjoint cycle form of , and so , as required.
In particular, a Sylow -subgroup of is self-normalizing. The following corollary is also immediate from Proposition 3.
Corollary 4. The number of Sylow -subgroups of is where .
Using the transitive action of on the vertices at each level of , it is not hard to prove the following lemma.
Lemma 5. The derived group of contains the permutations for all vertices at the same level of .
Corollary 6. The permutations for vertices at the same level of generate and .
Proof. Let be the vertex labelled at level of . Let be the subgroup generated by the permutations . By Lemma 5, is generated by the and . Moreover, the commute modulo . Therefore is abelian, and so , with the permutations generating the quotient group.
Weisner’s counting argument
In On the Sylow subgroups of the symmetric and alternating groups,
Amer. J. Math. 47 (1925), 121–124, Louis Weisner gives a counting argument that, specialized to , claims to show that has exactly Sylow -subgroups, where, as above . The exponent of is correct, but by Corollary 4, the exponent of is too large. Weisner’s error appears to be on page 123, where he implicitly assumes that Sylow subgroups of with the same base group are equal. This is false whenever and : for example
An alternative reference for the correct result is La structure des p-groupes de Sylow des groupes symétriques finis, Léo Kaloujnine, Ann. Sci. École Norm. Sup. (3) 65 (1948), 239–276: see page 262. In the footnote, Kaloujnine notes that, in general, the action of on does not induce the full automorphism group. Indeed, when and , the base group of the wreath product
is not even a characteristic subgroup of . For more on this see On the structure of standard wreath products of groups, Peter M. Neumann, Math. Z. 84 1964, 343–373.
The mistake in Weisner’s paper is pointed out on page 73 of an M.Sc. thesis by Sandra Covello.
Lower central series
The lower central series of was found by Weir in The Sylow subgroups of the symmetric groups, Proc. Amer. Math. Soc. 6 (1955) 534–541. Another description is given in the paper of Kaloujnine. There has been much further related work: see for example Lie algebra associated with the group of finitary automorphisms of p-adic tree, N. V. Bondarenko, C. K. Gupta V. I. Sushchansky, J. Algebra 324 (2010), 2198–2218 and the references to earlier papers therein. Using Lemma 6, one gets the description in Proposition 8 below. (This is almost certainly equivalent to a result in Kaloujnine’s paper stated using his formalism of tableaux polynomials, but the close connection with cyclic codes seems worth noting.)
Definition 7. Let be the th term in the lower central series of , so and for . Let for .
Let . Index coordinates of vectors in by the set labelling the vertices in . Given , we may define a corresponding element of by . This sets up a bijective correspondence between subgroups of and subspaces of . In turn, a subspace of invariant under the shift map sending to corresponds to an ideal in . (This is a basic observation in the theory of cyclic error-correcting codes.)
Proposition 8. We have
where corresponds to the ideal in generated by .
Proof. Under the correspondence above, multiplication by corresponds to conjugation by . Each is invariant under conjugation by , so we see that the subspaces of corresponding to the are all -invariant. Hence they are ideals in . It is easily seen by induction that is in the ideal corresponding to . Since the lower central series is strictly decreasing, and generates an ideal of codimension , the proposition follows.
Corollary 9. Let be the natural permutation module for , defined over a field of characteristic . Then the socle series for and the radical series for coincide; the th term (from the top) in either is the image of where .
Thus the radical and socle series of considered just as an -module coincide with the radical and socle series of as an -module. This is not really unexpected: the trivial module is the unique simple -module, and is uniserial even when considered as an -module. Still, it seems non-obvious without this injection of theory that is -invariant.
It is a nice exercise to find the decomposition of the analogously defined permutation module over : it has exactly summands of dimension for each and -dimensional summands, affording the characters of .
As a final application of the correspondence after Definition 7, we prove the following proposition.
Proposition 10. There are conjugacy classes of -cycles in . All -cycles in are conjugate.
Proof. Let . Let be the sum of the components of the vector corresponding to under the correspondence between and subspaces of set up after Definition 7. Standard results on conjugacy classes in wreath products (see for example Chapter 4 of The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley 1981) show that is a -cycle if and only if for each . Moreover, the conjugacy class of depends only on . By Proposition 3, the normalizer has a single orbit on this set.
Automorphism groups of cyclic codes
The previous section implies that the automorphism group of a -ary cyclic code of length contains a Sylow -subgroup of . In fact we get more: provided we label positions in the sensible way, so that the cyclic shift is , the automorphism group contains our distinguished Sylow subgroup . The results on above can be used to give a more powerful version of a result of Brand on equivalences of codes.
Proposition 11. Cyclic codes and of length are equivalent if and only if they are equivalent by an element of .
Proof. Let . We have seen that and . Hence so . By Sylow’s Theorem, there exists such that . Let . Clearly we have . Now apply Proposition 3 to .
For comparison, Brand’s result (see Lemma 3.1 in Polynomial isomorphisms of combinatorial objects, Graphs and Combinatorics 7 7–14) says that, for codes of arbitrary length, if is a Sylow -subgroup of , then the equivalence may be assumed to conjugate to another permutation in .
Application to Sylow’s Theorem
Let be a finite group of order . If acts transitively on a set of size coprime to then any point stabiliser contains a Sylow -subgroup of . The existence of Sylow subgroups of gives a very convenient setting to exploit this observation. By letting act regularly on itself, we may regard as a subgroup of . Fix a Sylow -subgroup of . The coset space has size coprime to . Let be in an orbit of of size coprime to and let be the stabiliser of . Now implies that is a subgroup of , so is a -subgroup of of order divisible by the highest power of dividing .
Intransitivity of derived group
By Corollary 6, any permutation in permutes amongst themselves the odd and even numbers in . Hence is not transitive. More generally, let be a transitive -group. By embedding in a Sylow -subgroup of , we see that is not transitive. Of course this can be proved in many other ways: for example, if is transitive then where is a point stabiliser. But , where is the Frattini subgroup of `non-generators’ of , so we have , a contradiction.
Automorphisms of permutation groups
Let be a transitive permutation group. It is natural to ask when every automorphism of is induced by the action of . For example, this is the case whenever acts regularly.
MAGMA code constructing all the permutations and groups defined in this post is available here.