Abstract Algebra homework Direct products

In summary, the conversation discusses the conditions needed for a group G to be isomorphic to the direct product of subgroups G_{1}, G_{2}, ..., G_{n}. These conditions include all subgroups being normal, elements of G being able to be written as a product of elements from the subgroups, and a specific intersection property. An example is given to show that if this intersection property is changed, G may not be isomorphic to the direct product of the subgroups. The use of homomorphisms and induction are suggested as possible approaches to proving this isomorphism.
  • #1
Avatarjoe
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Homework Statement



We've shown if G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{n}[/itex] are subgroups of G s.t.

1)G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{n}[/itex] are all normal
2)Every element of G can be written as g[itex]_{1}[/itex]g[itex]_{2}[/itex]...g[itex]_{n}[/itex] with g[itex]_{i}[/itex][itex]\in[/itex]G
3)For 1[itex]\leq[/itex]i[itex]\leq[/itex]n, G[itex]_{i}[/itex][itex]\cap[/itex]G[itex]_{1}[/itex],G[itex]_{2}[/itex],...,G[itex]_{i-1}[/itex]=e

then G[itex]\cong[/itex]G[itex]_{1}[/itex]xG[itex]_{2}[/itex]x...xG[itex]_{n}[/itex]

Show, by example, that if we replace 3) with the condition G[itex]_{i}[/itex][itex]\cap[/itex]G[itex]_{k}[/itex]=e for all i[itex]\neq[/itex]k then G does not need to be isomorphic to G[itex]_{1}[/itex]xG[itex]_{2}[/itex]x...xG[itex]_{n}[/itex]



Homework Equations





The Attempt at a Solution



I tried to find an example with abelian groups like[itex]Z[/itex][itex]_{60}[/itex], but nothing seemed to work. Now I'm trying groups that are themselves direct products, but I seem to be missing the big picture.
 
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  • #2
You need to find a good homomorphism

[tex]\varphi:G_1\times...\times G_n\rightarrow G[/tex]

and show that that is an isomorphism. What do you think you can choose as [itex]\varphi[/itex]??
 
  • #3
You also might want to think about an induction on n. Can you show it for n=2??
 

Related to Abstract Algebra homework Direct products

1. What is a direct product in abstract algebra?

A direct product in abstract algebra refers to the combination of two or more mathematical structures, such as groups, rings, or modules, into a new structure that shares some properties from each individual structure. It is denoted by the symbol × and has specific rules for combining elements and operations.

2. How do you find the direct product of two groups?

To find the direct product of two groups, G and H, you first create a new set G × H, where the elements are ordered pairs (g,h) with g∈G and h∈H. Then, you define a binary operation on the set as (g1,h1) × (g2,h2) = (g1g2, h1h2), where the operation on each component is the one defined in the corresponding group. This new set with the defined operation is the direct product of G and H.

3. What is the significance of direct products in abstract algebra?

Direct products play an important role in abstract algebra as they allow us to construct new structures with desired properties from existing ones. They also provide a way to study the relationship between different structures and understand their similarities and differences. Additionally, direct products have applications in various areas of mathematics, including number theory, geometry, and cryptography.

4. Is the direct product of two commutative groups also commutative?

No, the direct product of two commutative groups is not necessarily commutative. The commutativity of a direct product depends on the commutativity of the individual groups. If both groups are commutative, then the direct product will also be commutative. However, if one or both of the groups are non-commutative, then the direct product will not be commutative.

5. Can the direct product of two groups be isomorphic to one of the individual groups?

No, the direct product of two groups cannot be isomorphic to one of the individual groups. This is because the direct product combines the elements and operations of both groups, resulting in a new structure with different properties. Isomorphism requires that the two structures have the same underlying set and the same operation, which is not the case for the direct product.

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