Counting p-Sylow Subgroups for S4 and Z/5 with p=2, 3, and 5

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In summary, the number of p-Sylow subgroups for p=2, 3, and 5 are 2-Sylow: 0, 3-Sylow: 0, 5-Sylow: 1. The 2-Sylow subgroups of S4 are 3, while the 3-Sylow subgroups of S4 are 2. For Z/5, there are no 2-Sylow subgroups and no 3-Sylow subgroups, but there is 1 5-Sylow subgroup.
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Homework Statement


S4xZ/5

Find the number of p-Sylow subgroups for p=2, 3, and 5


The Attempt at a Solution



S4 is no problem. There I can use some binomial coefficients and count stuff. I was wondering how to tackle any Z/n.

2-Sylow

First we find 2-sylow subgroups of S4. [tex]\frac{\binom{4}{2}*2!}{2}[/tex] or 3.

Now I'm stuck, are the 2-Sylow subgroups of Z/5 just elements of order 2. I don't think there are any.
 
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3-SylowI can use the same binomial for 3-sylow subgroups of S4. \frac{\binom{4}{2}*2!}{3} or 2Now, I'm stuck again. Are the 3-sylow subgroups of Z/5 just elements of order 3. I don't think there are any. 5-SylowFor 5-Sylow subgroups of S4, I can use the same binomial. \frac{\binom{4}{2}*2!}{5} or 1The 5-sylow subgroups of Z/5 would just be elements of order 5, so there would be 1. So the answer is: 2-Sylow: 0, 3-Sylow: 0, 5-Sylow: 1
 

FAQ: Counting p-Sylow Subgroups for S4 and Z/5 with p=2, 3, and 5

How do you find the number of p-Sylow subgroups for S4 and Z/5 with p=2, 3, and 5?

The number of p-Sylow subgroups for a given group can be found using the Sylow theorems. For S4, the number of 2-Sylow subgroups is 6, the number of 3-Sylow subgroups is 4, and the number of 5-Sylow subgroups is 1. For Z/5, the number of 2-Sylow subgroups is 0, the number of 3-Sylow subgroups is 1, and the number of 5-Sylow subgroups is 1.

What is the order of the p-Sylow subgroups for S4 and Z/5 with p=2, 3, and 5?

The order of a p-Sylow subgroup is equal to p^k, where k is the largest power of p that divides the order of the group. For S4, the order of the 2-Sylow subgroup is 4, the order of the 3-Sylow subgroup is 3, and the order of the 5-Sylow subgroup is 5. For Z/5, the order of the 2-Sylow subgroup is 1, the order of the 3-Sylow subgroup is 3, and the order of the 5-Sylow subgroup is 5.

How are the p-Sylow subgroups determined for S4 and Z/5 with p=2, 3, and 5?

The p-Sylow subgroups are determined by finding all the elements of the group that have order equal to a power of p. For S4, the elements of order 2 form the 2-Sylow subgroups, the elements of order 3 form the 3-Sylow subgroups, and the identity element forms the 5-Sylow subgroup. For Z/5, the elements of order 3 form the 3-Sylow subgroup, and the identity element forms the 5-Sylow subgroup.

Can there be more than one p-Sylow subgroup for a given group?

According to the Sylow theorems, the number of p-Sylow subgroups for a given group must be a power of p. Therefore, if the order of the group is not a power of p, there can be multiple p-Sylow subgroups. In the case of S4 and Z/5, the number of p-Sylow subgroups is not a power of p, so there can be multiple p-Sylow subgroups.

How do the p-Sylow subgroups for S4 and Z/5 with p=2, 3, and 5 relate to each other?

For S4, the 2-Sylow subgroups and 3-Sylow subgroups are isomorphic to each other, meaning they have the same structure and properties. The 5-Sylow subgroup, on the other hand, is not isomorphic to the other two as it only contains the identity element. For Z/5, the 3-Sylow subgroup and 5-Sylow subgroup are both isomorphic to the group itself, while the 2-Sylow subgroup is trivial with only the identity element.

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