Describing the Sylow 2-subgroups of S5

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In summary, a Sylow 2-subgroup is a subgroup of a finite group with an order that is a power of 2. S5, the symmetric group on 5 elements, has 10 Sylow 2-subgroups. These subgroups can be described as the subgroups generated by 2-cycles and play an important role in the structure of the group. They have properties such as being self-normalizing, characteristic, and non-abelian.
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What is a good way to describe and count the Sylow 2-subgroups of S5? The ones isomorphic to D8 should be simple enough to describe as the extra conjugates of the Sylow 2-subgroups of S4, but I am trying to figure out how to easily describe any/all those isomorphic to Q8, which don't exist in S4.
 
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All Sylow p-subgroups of any group G are isomorphic (conjugate, in fact). Thus, if you can identify one copy of D8 in S5, you can immediately conclude that all Sylow 2-subgroups of S5 are isomorphic to D8.
 

1. What is a Sylow 2-subgroup?

A Sylow 2-subgroup is a subgroup of a finite group whose order is a power of 2. In other words, it is a subgroup that contains only elements whose order is a power of 2.

2. How many Sylow 2-subgroups does S5 have?

S5, which is the symmetric group on 5 elements, has 10 Sylow 2-subgroups. This can be calculated using the formula n2 = (n-1)!/2, where n is the order of the group.

3. How can the Sylow 2-subgroups of S5 be described?

The Sylow 2-subgroups of S5 can be described as the subgroups generated by 2-cycles, also known as transpositions. These subgroups have order 2 and are isomorphic to the cyclic group of order 2.

4. How are Sylow 2-subgroups related to S5's structure?

The Sylow 2-subgroups of S5 play an important role in the structure of the group. They are the building blocks of the normal subgroup A5, which has index 2 in S5. This means that every element in S5 can be written as a product of elements in the Sylow 2-subgroups and elements in A5.

5. What are the properties of the Sylow 2-subgroups of S5?

The Sylow 2-subgroups of S5 have several properties, such as being self-normalizing, meaning they are their own normalizers. They are also characteristic subgroups, meaning they are invariant under all automorphisms of S5. Additionally, they are non-abelian, as S5 is a non-abelian group.

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