ystael said:Do you know any general property of a group [tex]G[/tex] such that, when [tex]G[/tex] has this property and [tex]N[/tex] is a normal subgroup of [tex]G[/tex], you can conclude that [tex]G \cong N \times G/N[/tex]?
(Hint: in the direct product [tex]H \times K[/tex] of two groups, what is the relationship between the subgroups [tex]H \times 1[/tex] and [tex]1 \times K[/tex]?)
An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures that preserves their operations and structures.
A normal subgroup is a subgroup of a group that is closed under the group's operation and is also invariant under conjugation by elements of the group. In this context, it means that the subgroup is "compatible" with the larger group and can be used to construct an isomorphism between the two.
The isomorphism is constructed by mapping each element of G to a pair of elements, one from Z and one from Z_2, in a way that preserves the group operation. This mapping is possible because of the normal subgroup isomorphic to Z_2.
Yes, there can be multiple isomorphisms between two groups. In this case, there may be different ways to map the elements of G to elements of Z x Z_2 while preserving the group operation. However, all of these isomorphisms will be equivalent in the sense that they will have the same underlying structure.
The isomorphism between G and Z x Z_2 can be useful in solving problems and proving theorems in group theory. It allows us to understand the structure and properties of G by relating it to the more familiar group Z x Z_2. It also helps in constructing new groups by using the isomorphism to map elements between different groups.