Properties of the internal direct product of subgroups

In summary, the internal direct product of subgroups is a mathematical concept that refers to the product of two subgroups within a larger group, where the elements in both subgroups commute with each other and their intersection is only the identity element. It differs from the external direct product, which refers to the product of two groups as separate entities. The properties of the internal direct product include closure, associativity, commutativity, and the existence of identity and inverse elements. It is related to both the direct product and direct sum, and can be non-abelian if the subgroups do not commute with each other.
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bennyska
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Homework Statement


Let H and K be groups and let G = H x K. Recall that both H and K appear as subgroups of G in a natural way. Show that these subgroups H (actually H x {e}) and K (actually {e} x K) have the following properties:
a: every element of G is of the form hk for some h in H and k in K.
b: hk = kh for all h in H and k in K.
c: H INTERSECT K = {e}.


Homework Equations





The Attempt at a Solution


first off, I'm not sure what "Recall that both H and K appear as subgroups of G in a natural way" means. could someone explain that?
now, a seems easy enough. dealing with direct product of H and K defines elements of G as hk for some h and k in H and K respectively.
c, if I'm understanding it right, should actually be like {e} x {e}, since h looks like (h,e) and k looks like (e,k). these are both subgroups, so they both have e, and they would only coincide at {e} x {e}, or (e,e).
b, i have no idea. I'm not told G is abelian. the problem i did before this was to show that IF G is abelian, then the direct product is abelian, but... any clues?
 
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Thank you for your post. As a fellow scientist, I am happy to help you with your question.

To answer your first question, the statement "Recall that both H and K appear as subgroups of G in a natural way" simply means that H and K are already subgroups of G without any additional conditions or modifications. In other words, the group G is composed of the direct product of H and K, and H and K are not being forced into being subgroups of G.

For part a, you are correct in saying that every element of G is of the form hk for some h in H and k in K. This is because the direct product of H and K is defined as the set of all ordered pairs (h,k) where h is an element of H and k is an element of K. Therefore, every element of G can be written as (h,k) for some h and k in H and K respectively.

For part b, you are also correct that if G is abelian, then hk = kh for all h in H and k in K. However, even if G is not abelian, we can still show that hk = kh for all h in H and k in K. This is because the elements of H and K commute with each other, so when we take the direct product of H and K, the elements in H will still commute with the elements in K. Therefore, hk = kh for all h in H and k in K.

For part c, you are correct in saying that H ∩ K = {(e,e)}, or in other words, the only element that is common to both H and K is the identity element (e,e). This is because in the direct product of H and K, the elements are ordered pairs, so for an element to be in both H and K, it must be of the form (h,e) and (e,k) for some h in H and k in K. The only element that satisfies this is (e,e).

I hope this helps clarify the problem for you. Let me know if you have any further questions or need any additional explanations.
 

1. What is the internal direct product of subgroups?

The internal direct product of subgroups is a mathematical concept that refers to the product of two subgroups within a larger group, where the elements in both subgroups commute with each other and their intersection is only the identity element.

2. How is the internal direct product of subgroups different from the external direct product?

The internal direct product refers to the product of subgroups within a larger group, while the external direct product refers to the product of two groups as separate entities. In the internal direct product, the subgroups are already contained within the larger group, while in the external direct product, the groups are combined to form a new group.

3. What are the properties of the internal direct product of subgroups?

The properties of the internal direct product include closure, associativity, commutativity, and existence of identity and inverse elements. It also has the property that the elements in the subgroups commute with each other and their intersection is only the identity element.

4. How is the internal direct product related to the direct product and direct sum?

The internal direct product is a special case of both the direct product and direct sum. It is a direct product because the subgroups within the larger group commute with each other, and it is a direct sum because their intersection is only the identity element.

5. Can the internal direct product of subgroups be non-abelian?

Yes, the internal direct product can be non-abelian if the subgroups within the larger group do not commute with each other. However, it is always commutative if the subgroups commute with each other.

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