Question

(Modern Algebra) Show that every finite subgroup of the multiplicative group of a field is cyclical. (Hint: consider m as the order of the finite subgroup and analyze the roots of the polynomial (x ^ m) - 1 in field F)

Answer #1

Abstract Algebra (Modern Algebra)
Prove that every subgroup of an abelian group is abelian.

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.

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...

Give an example of a nontrivial subgroup of a multiplicative
group R× = {x ∈ R|x ̸= 0}
(1) of finite order
(2) of infinite order
Can R× contain an element of order 7?

(Modern Algebra) If G is a finite group with only two classes of
conjugation then the order of
G is 2.

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

Let G be a finite group, and suppose that H is normal subgroup
of G.
Show that, for every g ∈ G, the order of gH in G/H must divide
the order of g in G.
What is the order of the coset [4]42 +
〈[6]42〉 in Z42/〈[6]42〉?
Find an example to show that the order of gH in G/H does not
always determine the order of g in G. That is, find an example of a
group G, and...

Show that if H is a subgroup of index 2 in a finite group G,
then every left coset of H is also a right coset of H.
*** I have the answer but I am really looking for a thorough
explanation. Thanks!

Show that if G is a group, H a subgroup of G with |H| = n, and H
is the only subgroup of G of order n, then H is a normal subgroup
of G.
Hint: Show that aHa-1 is a subgroup of G
and is isomorphic to H for every a ∈ G.

True or False, explain:
1. Any polynomial f in Q[x] with deg(f)=3 and no roots in Q is
irreducible.
2. Any polynomial f in Q[x] with deg(f)-4 and no roots in Q is
irreducible.
3. Zx40 is isomorphic to
Zx5 x Zx8
4. If G is a finite group and H<G, then [G:H] = |G||H|
5. If [G:H]=2, then H is normal in G.
6. If G is a finite group and G<S28, then there is
a subgroup of G...

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