A class of non-abelian 2-groups containing Menon difference sets (Q1365497)

Let \(S_l\) be a 2-group with a chain of subgroups \(\{1\}\leq N_l\leq N_{l-1}\leq \cdots \leq N_0 \leq S_0 \leq S_1\leq\cdots \leq S_l\) such that, for each \(i\), the group \(N_i\) is elementary abelian of order \(2^{l+1-i}\), \(|S_i|=2^{l+2+i}\), and \(N_i\) is contained in the centre of \(S_i\). Then \(S_l\) contains a \((2^{2l+2},2^l(2^{l+1}-1),2^l(2^l-1))\) difference set. This is obtained using inductively a method of Meisner. Some examples show that the results are really different from previous similar results of Davis.