Question

Let I(G) denote the group of all inner automorphisms of a gorup G, and Aut(G) the group of all automorphisms of G. Show that I(G) is a normal subgroup of Aut(G).

Answer #1

8. Let g be an automorphism of the group G, and fa an
inner automorphism, as deﬁned
in Problems 3 and 4. Show that g ◦fa ◦g−1 is an inner automorphism.
Thus the group
of inner automorphisms of G is a normal subgroup of the group of
all automorphisms.

Let G = <a> be a cyclic group of order 12. Describe
explicitly all elements of Aut(G), the group of automorphisms of G.
Indicate how you know that these are elements of Aut(G) and that
these are the only elements of Aut(G).

Let H be a subgroup of G, and N be the normalizer of H in G and
C be the centralizer of H in G. Prove that C is normal in N and the
group N/C is isomorphic to a subgroup of Aut(H).

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 N and H be groups, and here for a homomorphism f:
H --> Aut(N) = group automorphism,
let N x_f H be the corresponding semi-direct product.
Let g be in Aut(N), and k be in Aut(H), Let C_g:
Aut(N) --> Aut(N) be given by
conjugation by g.
Now let z := C_g * f * k: H --> Aut(N), where *
means composition.
Show that there is an isomorphism
from Nx_f H to Nx_z H, which takes the natural...

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...

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

1) Let G be a group and N be a normal subgroup. Show that if G
is cyclic, then G/N is cyclic. Is the converse true?
2) What are the zero divisors of Z6?

Let G be a finite group and let P be a Sylow p-subgroup of G.
Suppose H is a normal subgroup of G. Prove that HP/H is a Sylow
p-subgroup of G/H and that H ∩ P is a Sylow p-subgroup of H. Hint:
Use the Second Isomorphism theorem.

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.

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