# Sylow's Counting Argument

• mathusers
In summary, the conversation discusses the use of Sylow's counting theorem to show that a group of order 44 must have a normal subgroup of order 11. This can then be used to classify all groups of order 44 by considering the possibilities of the number of Sylow 2-subgroups and using the fact that the Sylow 11-subgroup is cyclic and normal.

#### mathusers

Hi last one here. Any hints on this is appreciated too :)

Let G be a group of order 44. Show using Sylow's counting that G has a normal subgroup of order 11. Use the results to classify all groups of order 44.

How many subgroups of order 11 are there? (There are a bunch of other threads about Sylow counting arguments just like this one.)

ok here is what i have managed to get so far

By Sylow's 1st theorem, G has a subgroup of order 11. let $n_p$ be the number of sylow p-subgroups. then $n_{11} = 1(mod 11)$ and $n_{11}$ divides 2^2 (=4) so $n_{11}=1$. Therefore, it must be a normal subgroup (since it has no distinct conjugates).How can we use this to classify all groups of order 44 though?

Well, what are your thoughts? The obvious thing is to think about the Sylow 2s. There's either 1 or 11 of them (why?). If there's just 1, say H, then G is isomorphic to the direct product of H and the Sylow 11 (why?). Now what?

Scenario 2: there's 11 Sylow 2s. Take one of them and play around with it and with the Sylow 11. Remember that the Sylow 11 is cyclic and normal in G -- this will probably be useful in getting a presentation for G.

## 1. What is Sylow's Counting Argument?

Sylow's Counting Argument is a mathematical theorem developed by the Norwegian mathematician Ludwig Sylow. It is a powerful tool used to count the number of subgroups of a finite group, which helps in understanding the structure and properties of the group.

## 2. How does Sylow's Counting Argument work?

Sylow's Counting Argument involves using the concept of prime factorization to determine the number of subgroups of a finite group. It states that the number of subgroups of a group is equal to the product of its prime power divisors, where the prime powers are the highest powers of the prime numbers that divide the order of the group.

## 3. What are the applications of Sylow's Counting Argument?

Sylow's Counting Argument has various applications in different branches of mathematics such as group theory, number theory, and combinatorics. It is especially useful in proving the existence of certain subgroups and in studying the properties of finite groups.

## 4. Can Sylow's Counting Argument be used for infinite groups?

No, Sylow's Counting Argument is only applicable to finite groups. It relies on the concept of prime factorization, which is not defined for infinite numbers.

## 5. Are there any limitations to Sylow's Counting Argument?

Yes, Sylow's Counting Argument can only be used to count the number of subgroups of a finite group. It does not provide information about the specific structure or properties of the subgroups. Also, it is only applicable to groups with finite orders, so it cannot be used for infinite groups.