- #1
NasuSama
- 326
- 3
Homework Statement
Consider the group [itex]D_{4} = <x,y:x^2=1,y^4=1,yx=xy^3>[/itex] and the homomorphism [itex]\Phi : D_{4} \rightarrow Aut(D_{4})[/itex] defined by [itex]\Phi (g) = \phi _{g}[/itex], such that [itex]\phi _{g} = g^{-1}xg[/itex].
(a) Determine [itex]K = ker(\Phi)[/itex]
(b) Write down the cosets of [itex]K[/itex].
(c) Let [itex]Inn(D_{4}) = \Phi (D_{4})[/itex]. Then, [itex]\Phi : D_{4} \rightarrow Inn(D_{4})[/itex] is surjective. Exhibit the correspondence in the Correspondence Theorem explicitly.
Homework Equations
- Definition of the Correspondence Theorem
- Definition of the Kernel
- Definition of the Cosets
The Attempt at a Solution
I first list the elements in [itex]D_{4}[/itex], which are:
[itex]{e,x,y,y^2,y^3,xy,xy^2,xy^3}[/itex]
The order of [itex]D_{4}[/itex] is 8 since there are 8 distinct elements. My goal to answer the first problem is to find the elements in [itex]D_{4}[/itex] such that if I substitute the element from [itex]D_{4}[/itex] into [itex]\phi _{g}[/itex], then I obtain the identity. I would assume that the kernel of Φ consists of the cyclic groups of x², y^4 and x²y^4 since they yield the identity, and so substituting these elements for the function would give the identity. I may be wrong.
I didn't answer the second part since I need to get down the kernel of the whole function.
The third part might be related to the first two parts, but I'm not sure if the solutions are actually different.