Is direct product of subgroups a subgroup?

In summary, the conversation discusses a statement from a paper about the direct product of subgroups. The statement claims that the direct product of subgroups is automatically a subgroup, but one person does not understand how this can be true. They argue that one can take the direct product of a subgroup with itself multiple times and create a group of larger order than the parent group. However, the other person points out that the statement may have meant that the direct product of subgroups within the same group is a subgroup, not the direct product of subgroups from different groups.
  • #1
krishna mohan
117
0
Hi..

In the second paragraph of the following paper, there is a statement: "Because the direct product of subgroups is automatically a subgroup.."

http://jmp.aip.org/jmapaq/v23/i10/p1747_s1?bypassSSO=1 [Broken]

I don't see how that can be true...you can always take direct product of a subgroup with itself many times and create a group of order larger than the parent group...
 
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  • #2
If H_1 is a subgroup of G_1 and H_2 is a subgroup of G_2 then H_1 x H_2 is a subgroup of G_1 x G_2. In particular, if H1 and H2 are subgroups of a group G, then H1 x H2 is a subgroup of G x G (you are right, not of G itself).

I haven't read the article, but the authors might have meant this.
 

1. What is a direct product of subgroups?

A direct product of subgroups is a mathematical operation that combines two subgroups to form a new subgroup. It is denoted by H x K, where H and K are subgroups of a larger group G.

2. Is the direct product of subgroups always a subgroup?

Yes, the direct product of subgroups is always a subgroup as long as the two subgroups being combined are themselves subgroups of the larger group. This operation preserves the group structure and properties.

3. How is the direct product of subgroups different from the direct sum of subgroups?

The direct product and direct sum of subgroups are similar operations, but they differ in the way they combine the subgroups. The direct product combines the subgroups as a set, while the direct sum also takes into account the individual elements of the subgroups.

4. Can the direct product of subgroups be non-commutative?

Yes, the direct product of subgroups can be non-commutative. This means that the order in which the subgroups are combined matters and can result in different subgroups. However, the direct product is always associative, meaning that the order in which multiple direct products are performed does not matter.

5. How is the direct product of subgroups related to the Cartesian product of sets?

The direct product of subgroups is similar to the Cartesian product of sets, but with the additional condition that the elements in the direct product must satisfy the group operations of the larger group. The direct product of subgroups can also be seen as a generalization of the Cartesian product to groups.

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