Homework Help: Question about groups ?

1. Jul 28, 2011

cragar

1. The problem statement, all variables and given/known data
Just want to make something clear. Are all cyclic groups that have the same number of elements isomorphic to each other.

3. 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.

2. Jul 28, 2011

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.

3. Jul 28, 2011

cragar

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

4. Jul 28, 2011

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.

5. Jul 28, 2011