Question

Let p,q be prime numbers, not necessarily distinct. If a group G has order pq, prove...

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?

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

|G| = pq, where p & q are primes

If H is a Subgroup of G then, by Lagrange's theorem, |H| must divide |G| = pq

So, if, H is proper (and non-trivial) then, |H| = p or q, i.e. H Zp or H Zq

So, in both the cases, H is cyclic.

Even if H is trivial, i.e. H = {e} then also H is cyclic by default.

If, p = q, then, |G| = p² and |H| = p, so, H Zp

So, any proper Subgroup of G is cyclic.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose G is a group of order pq such p and q are primes, p<q and...
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.
Let G be a group of order p^2, where p is a prime. Show that G...
Let G be a group of order p^2, where p is a prime. Show that G must have a subgroup of order p. please show with notation if possible
if p and q are primes, show that every proper subgroup of a group of order...
if p and q are primes, show that every proper subgroup of a group of order pq is cyclic
Suppose that a cyclic group G has exactly three subgroups: G itself, e, and a subgroup...
Suppose that a cyclic group G has exactly three subgroups: G itself, e, and a subgroup of order p, where p is a prime greater than 2. Determine |G|
Let G be a finite group and let H, K be normal subgroups of G. If...
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].
Let p and q be any two distinct prime numbers and define the relation a R...
Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. For this relation R: Prove that R is an equivalence relation. you may use the following lemma: If p is prime and p|mn, then p|m or p|n
Let p and q be any two distinct prime numbers and define the relation a R...
Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. I need to prove that: a) R is an equivalence relation. (which I have) b) The equivalence classes of R correspond to the elements of  ℤpq. That is: [a] = [b] as equivalence classes of R if and only if [a] = [b] as elements of ℤpq I...
Let G be a group (not necessarily an Abelian group) of order 425. Prove that G...
Let G be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5. Note, Sylow Theorem is above us so we can't use it. We're up to Finite Orders. Thank you.
: (a) Let p be a prime, and let G be a finite Abelian group. Show...
: (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...
Consider all integers between 1 and pq where p and q are two distinct primes. We...
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?
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT