Find a p-Sylow subgroup for each of the given groups, and prime
p:
a. In Z24 a 2-sylow subgroup
b. In S4 a 2-sylow subgroup
c. In A4 a 3-sylow subgroup
(a) Let G and G′ be finite groups whose orders have no common
factors. Show that the only homomorphism φ:G→G′ is the trivial
one.
(b) Give an example of a nontrivial homomorphism φ for the given
groups, if an example exists. If no such homomorphism exists,
explain why.
i.φ: Z16→Z7
ii.φ: S4→S5
Let N be a normal subgroup of the group G.
(a) Show that every inner automorphism of G defines an
automorphism of N.
(b) Give an example of a group G with a normal subgroup N and an
automorphism of N that is not defined by an inner automorphism of
G
Let G be a nonabelian group of order 253=23(11), let P<G be a
Sylow 23-subgroup and Q<G a Sylow 11-subgroup.
a. What are the orders of P and Q. (Explain and include any
theorems used).
b. How many distinct conjugates of P and Q are there in G? n23?
n11? (Explain, include any theorems used).
c. Prove that G is isomorphic to the semidirect product of P and
Q.
1. Let N be a normal subgroup of G and let H be any subgroup
of G. Let HN = {hn|h ∈ H,n ∈
N}. Show that HN is a subgroup of G, and is the smallest
subgroup containing both N and H.
Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a
subgroup of G such that K ⊂ H Suppose that H is also a normal
subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b)
Show that G/H is isomorphic to (G/K)/(H/K).
4.- Show the solution:
a.- Let G be a group, H a subgroup of G and a∈G. Prove that the
coset aH has the same number of elements as H.
b.- Prove that if G is a finite group and a∈G, then |a| divides
|G|. Moreover, if |G| is prime then G is cyclic.
c.- Prove that every group is isomorphic to a group of
permutations.
SUBJECT: Abstract Algebra
(18,19,20)