Question

(8) Let G be a group of order of pq, where p and q are primes (p = q is allowed). (a) Show that every proper subgroup is cyclic. (b) Give an example that a group of order pq is not cyclic.

Name: (8) Let G be a group of order of pq, where p and q are primes (p = q is allowed). (a) Show that every proper subgroup is cyclic. (b) Give an example that a group of order pq is not cyclic.
Uploaded: 2023-09-02T08:27:35-08:00
Duration: 3 min 55 s
Channel: Adi S
Description: (8) Let G be a group of order of pq, where p and q are primes (p = q is allowed). (a) Show that every proper subgroup is cyclic. (b) Give an example that a group of order pq is not cyclic.

          (8) Let G be a group of order of pq, where p and q are primes (p = q is allowed). (a) Show that every proper subgroup is cyclic. (b) Give an example that a group of order pq is not cyclic.

(8) Let G be a group of order of pq, where p and q are primes (p = q is allowed). (a) Show that every proper subgroup is cyclic. (b) Give an example that a group of order pq is not cyclic.

Added by Shirley N.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/02/2023

Step 1

Now, suppose H is a proper subgroup of G. If H has order 1, then it is trivially cyclic. If H has order pq, then it is equal to G and is also cyclic. If H has order p or q, then by Lagrange's theorem, the index of H in G is either q or p, respectively. This Show more…

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Let G be a group of order of pq, where p and q are primes (p = q is allowed). (a) Show that every proper subgroup is cyclic. (b) Give an example that a group of order pq is not cyclic.