Dick said:A direct sum can be nonabelian. But why not if the order of the factor groups are 2 and 4?
Dickfore said:The question states direct product, not sum.
Dick said:Is there a difference when you are talking about groups? There is only one binary operation. That's a silly comment.
Dickfore said:What's a direct product of two groups and what's a direct sum?
Dick said:The distinction is unimportant. You only choose to say one or the other depending on whether you are using the additive notation for the group operation or the multiplicative. The result is the same. Your comments are not very helpful.
Dickfore said:Neither are yours.
D4, also known as the dihedral group of order 4, is a mathematical group that represents the symmetries of a square. It has 8 elements: 4 rotations and 4 reflections.
An internal direct product is a way of combining two smaller groups (subgroups) to create a larger group. Essentially, it means that the elements of D4 can be broken down into elements from two smaller groups in a specific way.
Proving that D4 cannot be expressed as an internal direct product is important because it helps us better understand the structure and properties of this group. It also has implications in other areas of mathematics, such as group theory and abstract algebra.
There are several ways to prove this, but one method is to show that D4 does not have any proper subgroups that are normal, nontrivial, and have elements of both rotations and reflections. This is a necessary condition for a group to be expressed as an internal direct product.
The proof has several implications, such as limiting the possible structures and properties of D4 and other similar groups. It also helps us classify and understand different types of groups, which has applications in various fields of mathematics and physics.