Isomorphic groups algebras with non-isomorphic groups

121 Views Asked by At

There is an interesting result of Dade about isomorphism of group algebras:

There exists finite non-isomorphic metabelian groups $G$ and $H$ whose group algebras over any field are isomorphic.

I was looking for the examples of these groups, but I didn't find them; only the result is stated in most of the books or slides online. So, I have one obvious question and one modification of it.

Question 1. What are the (counterexamples) groups constructed by Dade?

Question 2. Is the example by Dade, the smallest order example?

1

There are 1 best solutions below

0
On BEST ANSWER

You can find Dade's original paper at http://link.springer.com/article/10.1007/BF01109886.

He gives a construction of pairs of groups of order $p^3q^3$ for arbitrary primes $p,q$ with $q\equiv 1\pmod{p^2}$, so his smallest examples would be of order $1000$ ($p=2$ and $q=5$).

I've no idea whether smaller examples are possible using a different construction.