Are Cyclic Groups with the Same Order Isomorphic?

[cragar]

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.

 

[HallsofIvy]

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.

 

[cragar]

ok thanks for your answer, can we have non-cyclic groups be isomorphic to each other.

 

[stringy]

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 $x \mapsto e^x$.

However, remember that if G and G' are isomorphic groups, then G is cyclic if and only if G' is.

 

[cragar]

ok thanks for your answer