# Sylow Subgroups of Symmetric Groups

• Obraz35
In summary, the conversation was about finding a set of generators for a p-Sylow subgroup K of Sp2, as well as determining its order, normality, and whether it is abelian. The attempt at a solution involved finding the highest power of p that divides the order of the group and using that to determine the order of the Sylow p-subgroup. There was also discussion about trying examples with different primes to test the conjecture. The final idea proposed was to use powers of 9-cycles to generate the subgroup, but there was uncertainty about whether this approach would work.

## Homework Statement

Find a set of generators for a p-Sylow subgroup K of Sp2
.
Find the order of K and determine whether it is normal in Sp2 and if it is abelian.

## The Attempt at a Solution

So far I have that the order of Sp2 is p2!. So p2 is the highest power of p that divides the order of the group. Thus the Sylow p-subgroup has order p2 and because of that, has to be abelian. The other parts I'm not so sure on.

Are you sure you're right so far? Take p=2, S_4 has order 24, so its Sylow 2 subgroup has order 8. You seem to have assumed that p^2!=p!^2.

Sorry, it's supposed to be restricted to just odd primes, in which case I think that still holds, correct?

If you believe your conjecture, then it is the matter of no time at all to check if it's true for p=3 (and it obviously isn't). If you can't prove something in general try an example or 2.

Okay, that's true. Sorry.
I guess I really just have no idea how to find generators for this.
I was trying with p=3 for my example and found that 3^4 is the highest power of 3 that divides 9! since there are 4 factors of 3 in 9*8*7*6*5*4*3*2*1. So the Sylow p-subgroup would have 81 elements. So in general for odd prime p the order of the Sylow subgroup would be p^(p+1)?

I'm working on generators right now...

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Here is my idea although it may be way off the mark.
Still looking at p=3, the subgroup could contain powers of 10 different 9-cycles that way you'd have 80 different elements plus the identity. I think that works.

Although I guess we can't be sure that it's closed.

Well, I'm basically out of ideas now...

## What are Sylow Subgroups of Symmetric Groups?

Sylow subgroups are a type of subgroup that can be found in symmetric groups, which are groups of permutations or symmetries. They are named after mathematician Ludwig Sylow, who first studied and described them in the late 19th century.

## What is the significance of Sylow Subgroups?

Sylow subgroups have a special significance in group theory, as they can provide important information about the structure and properties of a group. They also have connections to other areas of mathematics, such as number theory and geometry.

## How can Sylow Subgroups be identified?

To identify Sylow subgroups in a symmetric group, one can use the Sylow theorems, which provide criteria for the existence and number of Sylow subgroups in a given group. These theorems involve concepts such as prime factorization and group order.

## What are the properties of Sylow Subgroups?

Sylow subgroups have several properties, such as being maximal subgroups and being conjugate to each other. They also have a specific order, which is a power of a prime number, and can be used to determine the order of the entire group.

## What is the role of Sylow Subgroups in group theory?

Sylow subgroups play a crucial role in group theory, as they can help classify and understand groups of symmetries. They also have applications in other areas of mathematics, such as Galois theory and the study of finite fields.