The discussion revolves around finding the order of the inner automorphisms of the dihedral group \(D_4\), which represents the symmetries of a square. Participants are exploring the relationship between \(Inn(D_4)\) and the center of \(D_4\), as well as the implications for the order of the group.
The discussion is ongoing, with participants raising questions about the center of \(D_4\) and its implications for the order of the inner automorphisms. Some guidance has been offered regarding the relationship between \(Inn(D_4)\) and \(D_4/Z(D_4)\), but there is no consensus on the exact nature of the groups involved or their representations.
Participants are working under the constraints of a homework problem, which may limit the information available for discussion. There is an emphasis on proving claims rather than assuming them, particularly regarding the structure of \(D_4\) and its center.
I think that the center is ##\{R_0, R_{180} \}##. So the order of ##D_4 / Z(D_4)## is 4, which makes the order of ##Inn( D_4)## 4?fresh_42 said:You've just proven ##Inn(D_4) \cong D_4/Z(D_4)##. So what is the center of ##D_4##?
Would have been my guess, too, but this is no proof. And even if we know the order were four, which one of the two groups of order four is it? do you know a representation of ##D_4## as a semi-direct product or which are the normal subgroups of ##D_4##?Mr Davis 97 said:I think that the center is ##\{R_0, R_{180} \}##. So the order of ##D_4 / Z(D_4)## is 4, which makes the order of ##Inn( D_4)## 4?