The discussion revolves around proving that the dihedral group D4 cannot be expressed as an internal direct product of two proper subgroups. The participants are exploring the properties of D4 and its subgroups, particularly focusing on the implications of non-commutativity.
The discussion is ongoing, with participants raising questions about the definitions and implications of direct products versus direct sums. There is a lack of consensus on the relevance of these distinctions, and the original poster is seeking further guidance on how to approach the proof.
Participants are navigating the definitions and properties of groups, particularly in relation to the specific structure of D4. The conversation reflects a mix of foundational understanding and clarification of terms, with no explicit resolution or agreement reached yet.
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.