# Product of two subgroups and intersection with p-subgroup

• moont14263
In summary, when considering a finite group G with a Sylow p-subgroup P, a normal subgroup K, and a subgroup H with (|K|,|H|)=1, if p divides |H|, then the intersection of P and HK is a subgroup of H. However, this may not hold true if K is not normal in G, as seen in the counterexample of S_3.
moont14263
Let G be a finite group. P is a Sylow p-subgroup of G and K is normal in G also H is a subgroup of G with (|K|,|H|)=1.
1) If p divides |H| then P$\cap$HK is a subgroup of H.
2) Is (1) when K is not normal in G.

This is my try of (1);
Let y be an element of P$\cap$HK, --> |y| divides |HK|=|H|*|K|--> |y| divides |H|--> y is an element of H as the order of y does not divide the order of K. What I am saying is that the elements of p power order of the intersection come from the Sylow p-subgroups of H.

For (2);
I know that |HK|=|H|*|K| but HK may not be a subgroup of G. And in this case also the intersection may not be a subgroup of G.

I can see that S_3 is a counter example for (1) and so there is no need to check (2). Thank you very much.

## 1. What is a "product of two subgroups"?

The product of two subgroups is a mathematical operation that combines elements from two different subgroups to form a new group. It is denoted as H1H2, where H1 and H2 are the two subgroups.

## 2. What is the significance of taking the intersection with a p-subgroup?

The intersection with a p-subgroup is important because it helps us understand the structure of the product of two subgroups. It allows us to identify elements that are common in both subgroups, which can reveal important information about the overall group.

## 3. Can the product of two subgroups and intersection with p-subgroup be commutative?

No, the product of two subgroups and intersection with p-subgroup is not generally commutative. This means that the order in which the subgroups are multiplied or intersected can affect the result.

## 4. How does the order of the subgroups affect the product and intersection?

The order of the subgroups does not affect the product, but it can affect the intersection. For example, H1H2 = H2H1, but H1∩H2 may not equal H2∩H1.

## 5. What are some real-world applications of studying the product of two subgroups and intersection with p-subgroup?

The product of two subgroups and intersection with p-subgroup has many applications in different fields of science, such as cryptography, chemistry, and computer science. It is also used in group theory to study the properties and structures of different groups.

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