Show that the groups of the following orders have a normal Sylow
subgroup.
(a) |G| = pq where p and q are
primes.
(b) |G| = paq where p and
q are primes and q < p.
(c) |G| = 4p where p is a prime greater than
four.
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
Use the classification of groups with six elements to show that
A(4) has no subgroup with 6 elements. [ Hint: check that the
product of any two elements of A(4) of order 2 has order 2]
Investigate the following theorems
(h) For sets A, B and C we have
i. A\(B ∪ C) = (A\B) ∩ (A\C),
ii. A\(B ∩ C) = (A\B) ∪ (A\C),
iii. A ̸= B if and only if (A\B) ∪ (B\A) ̸= ∅,
iv. A ∪ B ⊆ C if and only if A ⊆ C and B ⊆ C.
What happens in the extreme case(s) where some (or all) sets are
empty?