(a) Let G be an abelian group of order 2020. Prove that G has a subgroup that is isomorphic to Z/1010Z. Hint. You may use Exercise 5.12 (b) Give an example of an abelian group G of order 2020 that is not isomorphic to Z/2020Z (also prove that indeed G ? Z/2020Z).
Added by Michelle D.
Close
Step 1
12, we know that any abelian group of order 2020 can be written as a direct product of cyclic groups of prime power order. Since 2020 = 2^2 * 5 * 101, the group G can be expressed as G = Z/2^2Z x Z/5Z x Z/101Z. ** Show more…
Show all steps
Your feedback will help us improve your experience
Adi S and 87 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
(a) Show that a group that has only a finite number of subgroups must be a finite group. (b) Let G be a group that has exactly one nontrivial, proper subgroup. Show that G must be isomorphic to Zp2 for some prime number p. (Hint: use part (a) to conclude that G is finite. Let H be the one nontrivial, proper subgroup of G. Start by showing that G and hence H must be cyclic.)
Hoan N.
Math: Number Theory Q. Show that up to isomorphism, the only groups of order 4 are Z / 4Z and Z / 2Z × Z / 2Z. Give examples of 3 nonisomorphic groups of order 8. Show that they are not isomorphic to each other. Hint: Study the multiplication table
Sri K.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD