Can a Sylow Subgroup be Contained in a Sylow p-Subgroup?

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In summary, the problem states that if J is a subgroup of G with an order that is a power of a prime p, then J must be contained in a Sylow p-subgroup of G. This is based on a lemma that states a subgroup H is normal if every orbit of its induced action on a coset contains only one coset. In this problem, H is taken to be a Sylow p-subgroup and the induced action of J on X is considered. When there is only one Sylow p-subgroup, the lemma can be used. However, when there are multiple Sylow p-subgroups, the converse of the theorem that any two Sylow p-subgroups are conjugate may be
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Homework Statement


If J is a subgroup of G whose order is a power of a prime p, verify that J must be contained in a Sylow p-subgroup of G. The problem says to refer to a lemma that given an action by a subgroup H on its own left cosets, h(xH)=hxH, H is a normal subgroup iff every orbit of the induced action of H on a coset contains just one coset. In the problem it says to take H in the lemma to be a Sylow p-subgroup and consider the induced action of J on X.

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The Attempt at a Solution


I'm really not sure how to do this. I've been trying to start with the case that there's only one Sylow p-subgroup (H). Then I know that it's normal, so I could ideally use the lemma. I don't know how though, and I'm totally baffled by the case where there's more than one Sylow p-subgroup. A fellow student was trying to use the theorem that any two Sylow p-subgroups are conjugate, to simplify the case where there's more than one. But to me it seems like you'd need the converse of that theorem to get anywhere with it.
 
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If p is a prime where the order of G is 2p, then G has a normal subgroup, J, with order p (if p is not equal to 2).

Since J is normal, then it is contained in the intersection of all Sylow p-subgroups in G (I thought this was by definition... I don't know). Thus, it must be contained in at least one Sylow p-group of G.

Perhaps someone else can help you with this. Cheers.
 

What is a Sylow subgroup?

A Sylow subgroup is a subgroup of a finite group that has a prime power order and is as large as possible, meaning it is a maximal subgroup of that particular order.

How are Sylow subgroups related to normal subgroups?

Sylow subgroups are not always normal subgroups, but they can be used to determine the number of normal subgroups in a finite group. In particular, the number of Sylow subgroups of a certain order is equal to the index of the normalizer of that subgroup, which can be used to determine the number of normal subgroups of that order.

Can a group have more than one Sylow subgroup of the same order?

Yes, a group can have more than one Sylow subgroup of the same order. In fact, if a group has more than one Sylow subgroup, they are all conjugate to each other and thus are isomorphic. This is known as the Sylow’s third theorem.

How are Sylow subgroups useful in group theory?

Sylow subgroups play an important role in the classification of finite groups. By analyzing the properties of Sylow subgroups, we can determine the structure and properties of a finite group. They are also useful in proving other theorems and results in group theory.

Can every finite group have a Sylow subgroup?

Yes, every finite group has at least one Sylow subgroup. This is known as the Sylow’s first theorem. However, it is not necessary that every finite group has Sylow subgroups of every order. For example, a prime order group has only one Sylow subgroup, which is the group itself.

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