I seek to find the kernel of the homomorphism $$\theta(g) = A^{-1}gA$$ from the dihedral group $$D_4$$ of order $8$ to itself. Where A is a fixed element of $$D_4$$
I know that the kernel consists of the element $g$ is such that $$A^{-1}gA =1$$ which would imply that $$g=AA^{-1}$$. Is this a specific group?
I also wanted to prove that it is injective so I took $$g_1, g_2$$ such That $$A^{-1}g_1A= A^{-1}g_2A$$ if I multiply by A on the left side and A inverse on the right I gEt that $$g_1 = g_2 $$ Is this correct?