From Wikimedia Commons, the free media repository. Subcategories This category has the following 17 subcategories, out of 17 total. Media in category "Abstract algebra" The following 78 files are in this category, out of 78 total. Algebraic structures.
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In this section, we will have a look at the Sylow theorems and their applications. The proofs are a bit difficult but nonetheless interesting. Important remark: Wikipedia also has proofs of the Sylow theorems, see Wikipedia article on the Sylow theorems , which are shorter and more elegant. But here you can find other proofs. This is because the author wanted to avoid redundancy. Definition 2 : Let H be a subgroup of a group G.
We define the normalizer N[H] of H as follows:. Then there exists an element of G which has order p. Proof : For this proof, we use induction. Let H be a p-subgroup of G, i. But also the following equivalences are true:. P acts on X by conjugation. Now we let Q act on X by conjugation, instead of P. So what does this mean? Then the normalizer of H is the stabilizer of H in this action. Therefore, it is a subgroup due to Lemma 14 of the section about group actions, QED.
Proof : This follows from the proof of Sylow II and the thm. But since due to the orbit-stabilizer theorem thm. This proves the second part. In this section, it will be shown how to show that groups of a certain order can not be simple using the Sylow theorems. This is a useful application of the Sylow theorems. But since, due to Sylow II, the conjugate of the only Sylow 5-group is again a Sylow 5-group, it is itself.
This is the definition of normal subgroups. Therefore, by the definition of simple groups, groups of order are not simple.
This is due to theorem 2 from the section about group actions. The image can't be trivial, because all Sylow 2-groups are conjugate because of Sylow II. We let G act on the cosets of S by left multiplication. Therefore, the kernel is a proper, non-trivial normal subgroup, which is why the group is not simple. From Wikibooks, open books for an open world.
Lemma 8 : The normalizer of a subgroup is a subgroup. How to show that groups of a certain order aren't simple [ edit ] In this section, it will be shown how to show that groups of a certain order can not be simple using the Sylow theorems. Example 11 : Groups of order are not simple.
Example 12 : Groups of order 48 are not simple. Example 13 : Groups of order are not simple. Category : Book:Abstract Algebra. Namespaces Book Discussion. Views Read Edit View history.
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