Isoclinism of groups

Do you know when two groups \(G\) and \(H\) are called isoclinic to each other? I have never heard of this notion before, but there was a paper on the arXiv listing today using it in its abstract; so I looked in this paper, not because I was interested in its content, but just because I was interested what this notion is.

For a group \(G\) we consider the following data: The quotient \(G/Z(G)\) of \(G\) by its center, the commutator subgroup \([G,G]\) of \(G\), and the commutator map \(G/Z(G) \times G/Z(G) \to [G,G]\) induced by \((a,b) \mapsto aba^{-1}b^{-1}\).

The groups \(G\) and \(H\) are now called isoclinic to each other, if there are isomorphisms \(G/Z(G) \to H/Z(H)\) and \([G,G] \to [H,H]\) intertwining the commutator maps.

Actually, this notion even has a wikipedia entry: link.

Leave a Reply

Your email address will not be published.