Existence of weird automorphisms in finite groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:39:04Z http://mathoverflow.net/feeds/question/59617 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59617/existence-of-weird-automorphisms-in-finite-groups Existence of weird automorphisms in finite groups Makhalan Duff 2011-03-25T22:44:12Z 2011-03-25T22:44:12Z <p>I've been contemplating various conjectures on various fields that a priori don't have anything to do with group theory; and yet these heuristics (that it would take too long to go over, and are besides the point here) seem to imply the following very weird proposition:</p> <p>Fix a prime $p\geq 3$. Let $G$ be a finite, prime-to-$p$, group generated by $r$ elements: $a_1,...,a_r$. Assume $r\leq p$. Then there exists an automorphism $\phi \in Aut(G)$ such that $G/$the group normally generated by $a_i \phi(a_i)^{-1}$ (for every $i$) is generated by elements whose order divides $p-1$.</p> <p>This is true for $r=1$ (pick $\phi(a_1)=a_1^{-1}$).</p> <p>I can prove it for some easy examples. Can you think of a reason for this to be true/of a counterexample?</p>