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Sylow p-subgroup

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Definition

In a finite group GG, for every prime pp, a maximal pp-torsion subgroup of GG is also known as a Sylow pp-subgroup.

Facts

Let the order of GG be rp k r p^k , where r r is coprime to pp.

Let H<GH \lt G be a subgroup of rank p lp^l, and consider the left action of GG on right cosets of HH:

(g,xH)(gx)H. (g, x H )\mapsto (g x) H .

This action induces further an action of GG on (G/H)C(p kl)(G / H) \mathbf{C} (p^{k-l}), the subsets of G/HG/H of size p klp^{k-l}. The number of these, (rp klp kl)\binom{ r p^{k-l} }{ p^{k-l} } , is congruent to rr, mod pp, and hence some orbit must have size coprime to pp, hence necessarily dividing rr, hence some set of cosets must have stabilizer of size at least p k p^k . One checks that on the other hand, the stabilizer of a set of cosets is at most the size of their union for a very good reason, and furthermore is a subgroup of GG. Lastly, every orbit contains a representative that contains HH. In consequence,

Theorem

Every pp-subgroup HH of GG is contained in a subgroup of order p kp^k, which is necessarily a maximal pp-subgroup. The number of maximal pp-subgroups including HH is congruent to 11 mod pp.

One also has

Theorem

Any two Sylow pp-subgroups of GG are conjugate.

See class equation for a detailed discussion of these matters. (Now updated to take into account the proof below. The discussion above refers to a more involved proof from an earlier page version, which in turn was adapted from the Wikipedia article; it may be found here, comment 4.) The following slick proof for the existence of Sylow subgroups was suggested to us by Benjamin Steinberg.

Proof that Sylow subgroups exist

First observe that if a group GG has a pp-Sylow subgroup PP, then so does each of its subgroups HH. For we let HH act on G/PG/P by left translation, and then note that since G/PG/P has cardinality prime to pp, so must one of its connected components H/Stab(a x)H/Stab(a_x) in the HH-set decomposition

G/P orbitsxH/Stab(a x)G/P \cong \sum_{orbits\; x} H/Stab(a_x)

(a xG/Pa_x \in G/P a representative of its orbit xx), making Stab(a x)Stab(a_x) a pp-Sylow subgroup of HH.

Then, if HH is any group, apply this observation to the embedding

HCayleyPerm(|H|)GL |H|(/(p))H \stackrel{Cayley}{\hookrightarrow} Perm({|H|}) \hookrightarrow GL_{{|H|}}(\mathbb{Z}/(p))

where we embed the permutation group via permutation matrices into the group GG consisting of matrices |H|×|H|/(p){|H|} \times {|H|} \to \mathbb{Z}/(p). Letting nn be the cardinality of HH, the order of GG is (p n1)(p np)(p np n1)(p^n - 1)(p^n - p)\ldots (p^n - p^{n-1}), with maximal pp-factor p n(n1)/2p^{n(n-1)/2}. This GG has a pp-Sylow subgroup given by unitriangular matrices, i.e., upper-triangular matrices with all 11‘s on the diagonal, and we are done.

References

For a generalisation of Sylow theory to finite ∞-groups, that is, ∞-groups with finitely many non-trivial homotopy groups which are all finite, see

Last revised on September 26, 2018 at 17:43:23. See the history of this page for a list of all contributions to it.