I am studying an undergraduate course on group theory, and I have a question about the following remark made in the lecture notes:
The internal direct products are useful for helping us sort out the structure of a given group. External direct products are useful for constructing new groups from old groups.
I understand that the internal and external direct products are essentially the same thing (at least I think that's the case).
I can't find any examples where, given a group, simply our knowledge of the internal direct product helps us understand the structure of that group, unless we already know the subgroups which our group is the internal direct product of.
Does the quote above imply that we can use only our knowledge of the internal direct product to understand the structure of a group?
Or, does it actually mean that if the subgroups (which the group is the internal direct product of) are known, then we can understand the structure of the group?