Nikolaus Conference 2022

Let \(p\) be a prime and \(P\) be a finite \(p\)-group. We define a new characteristic subgroup \(\textbf{K}(P)\) of \(P\) and show that \(\textbf{K}(P)\) is a critical subgroup of \(P\), which improves Thompson's proof for their existence. Secondly, we define several new characteristic subgroups \(\textbf{K}^*(P)\), \(\textbf{K}_i(P)\) for each positive integer \(i\) and show that they enjoy some interesting and new properties similar to the those of critical subgroups. Lastly, for a solvable group \(G\), we define a new characteristic subgroup \(\textbf{Y}(G)\), which satisfies the analogue of Thompson's result for solvable groups.