The Sylow theorems
ثبت نشده
چکیده
Lagrange’s theorem tells us that if G is a finite group and H ≤ G, then #(H) divides #(G). As we have seen, the converse to Lagrange’s theorem is false in general: if G is a finite group of order n and d divides n, then there need not exist a subgroup of G whose order is d. The Sylow theorems say that such a subgroup exists in one special but very important case: when d is the largest power of a prime which divides n. (It then turns out that G has a subgroup of every order which is a prime power dividing n, not necessarily the largest such.) In fact, the Sylow theorems tell us much more about such subgroups, by giving information on how many such subgroups can exist. As we shall see, this will sometimes enable us to show that G has a nontrivial proper normal subgroup.
منابع مشابه
Group Actions, p-Groups, and the Sylow Theorems
In this note we introduce the notion of a group action on a set and use it to prove a number of theorems about p-groups and the Sylow theorems. For undefined terms see any decent book on group theory. The theory of p-groups and the Sylow theorems have a number of applications in Galoistheory. The nice structure of p-groups will translate via the fundamental theorem of Galois theory to nice stru...
متن کاملSecondary Sylow Theorems
These theories extend the existent proof of the first sylow theorem (written by Florian Kammueller and L. C. Paulson) by what is often called the second, third and fourth sylow theorem. These theorems state propositions about the number of Sylow p-subgroups of a group and the fact that they are conjugate to each other. The proofs make use of an implementation of group actions and their properties.
متن کاملAlgebra Prelim Fall 2013
First we will prove a small lemma. Lemma 1. Let P be a p-Sylow subgroup of G. Let N be a normal subgroup of G such that P ⊂ N . Then all p-Sylow subgroups are in N . Proof. Let P̃ be a p-Sylow subgroup that is not P . Then by the Sylow Theorems, we know that gPg−1 = P̃ for some g ∈ G. Since P ⊂ N , and N is normal, we know that gPg−1 ⊂ N and hence P̃ ⊂ N . Thus all p-Sylow subgroups are in N . Now...
متن کاملFormalising Sylow's theorems in Coq
This report presents a formalisation of Sylow’s theorems done in Coq. The formalisation has been done in a couple of weeks on top of Georges Gonthier’s ssreflect [2]. There were two ideas behind formalising Sylow’s theorems. The first one was to get familiar with Georges way of doing proofs. The second one was to contribute to the collective effort to formalise a large subset of group theory in...
متن کاملCourse 311: Michaelmas Term 2005 Part II: Topics in Group Theory
2 Topics in Group Theory 2 2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Examples of Groups . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Elementary Properties of Groups . . . . . . . . . . . . . . . . 4 2.4 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Co...
متن کامل