Are Cyclic Groups with the Same Order Isomorphic?

  • Thread starter Thread starter cragar
  • Start date Start date
  • Tags Tags
    Groups
Click For Summary

Homework Help Overview

The discussion revolves around the properties of cyclic groups, specifically whether all cyclic groups of the same order are isomorphic. Participants explore the implications of group isomorphism and the characteristics of cyclic versus non-cyclic groups.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants assert that cyclic groups of the same order are isomorphic, citing one-to-one correspondence and the presence of generators. Others question whether non-cyclic groups can also be isomorphic, leading to a broader exploration of group properties.

Discussion Status

The discussion includes affirmations regarding cyclic groups and their isomorphisms, with some participants providing examples and counterexamples. There is an ongoing exploration of the relationship between cyclic and non-cyclic groups, with no explicit consensus reached.

Contextual Notes

Participants are considering the definitions and properties of group isomorphism, particularly in the context of cyclic and non-cyclic groups. The discussion reflects a mix of assumptions and clarifications regarding these concepts.

cragar
Messages
2,546
Reaction score
3

Homework Statement


Just want to make something clear. Are all cyclic groups that have the same number of elements isomorphic to each other.

The Attempt at a Solution


I think yes because theirs is a one-to-one correspondence and the groups are cyclic which means they have generators.
 
Physics news on Phys.org
Yes, all cyclic groups of order n are isomorphic. You can define the isomorphism itself by mapping the generator of one group to the generator of the other group.
 
ok thanks for your answer, can we have non-cyclic groups be isomorphic to each other.
 
Does an isomorphism between groups require that the groups be cyclic? No. Take the additive group of real numbers and the multiplicative group of positive reals. They are isomorphic via [itex]x \mapsto e^x[/itex].

However, remember that if G and G' are isomorphic groups, then G is cyclic if and only if G' is.
 
ok thanks for your answer
 

Similar threads

Proving that an Abelian group of order pq is isomorphic to Z_pq
  • · Replies 5 ·
Replies
5
Views
3K
Characterizing subgroups of a cyclic group
  • · Replies 9 ·
Replies
9
Views
2K
Showing that cyclic groups of the same order are isomorphic
  • · Replies 3 ·
Replies
3
Views
2K
If G is cyclic, and G is isomorphic to G', then G' is cycli
  • · Replies 1 ·
Replies
1
Views
2K
Is the group of positive rational numbers under * cyclic?
  • · Replies 3 ·
Replies
3
Views
4K
Group is a union of proper subgroups iff. it is non-cyclic
  • · Replies 1 ·
Replies
1
Views
5K
Show the group of units in Z_10 is a cyclic group of order 4
  • · Replies 3 ·
Replies
3
Views
2K
Every infinite cyclic group has non-trivial proper subgroups
  • · Replies 3 ·
Replies
3
Views
2K
Calculating a pretty simple Galois Group
  • · Replies 19 ·
Replies
19
Views
2K
Abelian group as a direct product of cyclic groups
  • · Replies 9 ·
Replies
9
Views
2K