Question

Let Hi be a subgroup of G that is not normal in G. Let H-ф-1H1ф be a cong gate subgroup. (i) ф is an automorphism of F. Show that its restricts to an isomorphism ф : FH2-> FHI. (iüi) Show that if a e Fla but not in Flh n Fta, and if f is the irreducible polynomial for a, then f does not split over FHa (thus Fs is not a normal extension).

Answer 1

i) Let $x\in F^{H_2}$; then for any
h E H1
we have
0-1 (h(c)(z))) r=
and therefore, $\phi(x)=h(\phi(x))$ . Hence, $\phi(x)\in F^{H_1}$ . Thus, $\phi:F^{H_2}\rightarrow F^{H_1}$ is well-defined. Being a restriction of an automorphism, $\phi:F^{H_2}\rightarrow F^{H_1}$ is a homomorphism and injective.

Let
EFH
and let
TO EF
be such that $\phi(x_0)=y$; then for any
h E H1
we have

$\phi^{-1}(h(\phi(x_0)))=\phi^{-1}(h(y))=\phi^{-1}(y)=\phi^{-1}(\phi(x_0))=x_0$

Hence, $x_0\in F^{H_2}$, showing that $\phi:F^{H_2}\rightarrow F^{H_1}$ is surjective, and hence, an isomorphism.

ii) If possible suppose that splits in
EH
. Since
o(f) = f
, we have
BoB
for all zero $\beta\in F^{H_2}$ of . Thus, $\phi:F^{H_2}\rightarrow F^{H_1}$ maps the set of zeros of to itself, and this set of zeros lies in $F^{H_2}\cap F^{H_1}$ . Since $\alpha\in F^{H_2}$ but $\alpha\notin F^{H_2}\cap F^{H_1}$ , this is a contradiction. Hence, does not split in
EH
.