# center, centralizer

• November 9th 2010, 03:10 PM
kathrynmath
center, centralizer
Let D4 = {e, r, r2, r3, f, fr, fr2, fr3}, where r4 = f2 = e and rf = fr−1 = fr3.
(a) Find the centralizer CD4(r) of r and the centralizer CD4(f) of f in D4.
(b) Find the center Z(D4) of D4.

The centralize is C(a) with all elements of D4 that commute with a.
We want xr=rx
C(r)={e,r,r^2,r^3,f}
want xf=fx
C(f)={e,r,f,fr^2}

center is elements that commute with every other element of D4 z(D4)
want xg=gx
Z(D4)={e,r,f}

not sure if any of the values are correct or if I'm missing some.
• November 9th 2010, 05:53 PM
Drexel28
Quote:

Originally Posted by kathrynmath
Let D4 = {e, r, r2, r3, f, fr, fr2, fr3}, where r4 = f2 = e and rf = fr−1 = fr3.
(a) Find the centralizer CD4(r) of r and the centralizer CD4(f) of f in D4.
(b) Find the center Z(D4) of D4.

The centralize is C(a) with all elements of D4 that commute with a.
We want xr=rx
C(r)={e,r,r^2,r^3,f}
want xf=fx
C(f)={e,r,f,fr^2}

center is elements that commute with every other element of D4 z(D4)
want xg=gx
Z(D4)={e,r,f}

not sure if any of the values are correct or if I'm missing some.

I'm fairly sure that the rule is that if $n>2$ and $2\mid n$ that, with your notation, $\mathcal{Z}\left(D_{2n}\right)=\left\langle r^{\frac{n}{2}}\right\rangle$. So, I'd say that you're answer is probably wrong.

That said, you haven't shown any work we could judge?