Question

Prove that if F is a field and K = F^{G} for a finite
group G of automorphisms of F, then there are only finitely many
subfields between F and K.

Please help!

Answer #1

4. Let f : G→H be a group homomorphism. Suppose a∈G is an
element of finite order n.
(a) Prove that f(a) has finite order k, where k is a divisor of
n.
(b) If f is an isomorphism, prove that k=n.

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Let G be a finite group and let H, K be normal subgroups of G.
If [G : H] = p and [G : K] = q where p and q are distinct primes,
prove that pq divides [G : H ∩ K].

Let F be a field. It is a general fact that a finite subgroup G
of (F^*,X) of the multiplicative group of a field must be cyclic.
Give a proof by example in the case when |G| = 100.

If R is a UFD and f(x), g(x) contains R[x], prove that c(fg) and
c(f)c(g) are associatives.

Let E/F be a finite Galois extension such that Gal(E/F) is
abelian. Prove that for
every intermediate field K, the extension K/F is Galois.

Prove that if (G, ·) is a finite group of even order, then
there always exists an element g∈G such that g ≠ 1 and
g2=1.

True/False, explain:
1. If G is a finite group and G28, then there is a subgroup of G of
order 2401=74
2. If |G|=19, then G is isomorphic to Z19.
3. If F subset of K is a degree 5 field extension, any element b in
K is the root of some polynomial p(x) in F[x]
4. If F subset of K is a degree 5 field extension, viewing K as
a vector space over F, Aut(K, F) consists of...

Let G be a finitely generated group, and let H be normal
subgroup of G. Prove that G/H is finitely generated

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

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