# Group Theory question

1. Nov 15, 2008

### herbert_454

Dear All,

Is it true that one can find some 10 groups (from different isomorphism classes) with order between (and including) 25 and 29 such that each pair of the same order are not isomorphic to each other? If so, how does one go about generating such a list and showing they are not isomorphic to one another?

Best,
Herbert

2. Nov 15, 2008

### morphism

This should be easy to google.

Thinking out loud.. There's only one of order 25=5^2 (cyclic). Only two of order 26=2*13 (cyclic and dihedral). There are 3 abelian groups of order 27, 2 abelians and a dihedral of order 28 and only one group of order 29. So that's 10 groups right there. Any remaining groups will have to have order 27 or 28. I'll let you figure out if there are any more left.

3. Nov 16, 2008

### HallsofIvy

Staff Emeritus
Morphism, why would there be only the cyclic group of order 25? It's not just because the order is a prime squared because there are two non-isomorphic groups of order 4= 22, the cyclic group and the Klein four group.

4. Nov 16, 2008

### herbert_454

Thanks! I got it. Yes there is another order 25 group, namely C5 x C5 :)

5. Nov 16, 2008

### morphism

Oops..! What I should have said was that a group of order 25 is abelian - because its order is prime squared. I took it one step too far! Thanks for spotting that Halls.