Question

Suppose G is a group of order pq such p and q are primes, p<q
and therefore |H|=p and |K|= q, where H and K are proper subgroups
are G. It was determined that H and K are abelian and G=HK. Show
that H and K are normal subgroups of G **without using
Sylow's Theorem**.

Answer #1

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].

I want to proove
Q) Any group of order pq (where p,q are primes) is solvable

Let p,q be prime numbers, not necessarily distinct. If a group G
has order pq, prove that any proper subgroup (meaning a subgroup
not equal to G itself) must be cyclic. Hint: what are the possible
sizes of the subgroups?

Let H and K be subgroups of a group G so that HK is also a
subgroup. Show that HK = KH.

Consider all integers between 1 and pq where p and q are two
distinct primes. We choose one of them, all with equal
probability.
a) What is the probability that we choose any given number?
b) What is the probability that we choose a number that is
i) relatively prime to p?
ii) relatively prime to q?
iii) relatively prime to pq?

A group G is a simple group if the only normal subgroups of G
are G itself and {e}. In other words, G is simple if G has no
non-trivial proper normal subgroups.
Algebraists have proven (using more advanced techniques than
ones we’ve discussed) that An is a simple group for n ≥ 5.
Using this fact, prove that for n ≥ 5, An has no subgroup of
order n!/4 .
(This generalizes HW5,#3 as well as our counterexample from...

Suppose that G is a group with subgroups K ≤
H ≤ G. Suppose that K is normal in
G. Let G act on G/H, the set of
left cosets of H, by left multiplication. Prove that if k
∈ K, then left multiplication of G/H by
k is the identity permutation on G/H.

: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

Suppose that n is a product of two k-bit primes p and q. Suppose
also that it is known that
|p-q|<2t, where t is small. DESCRIBE a way to find the
factorization of n in t steps. (Note: in terms of RSA, it shows
that although we want p and q to be of similar size, it is also
undesirable that p and q are very close)

If G is a group of order (p^k)s where p is a prime number such
that (p,s)=1, then show that each subgroup of order p^i ; i=
1,2...(k-1) is a normal subgroup of atleast one subgroup of order
p^(i+1)

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