Applications of Commutator-Type Operators in p-Groups.

G'abor Luk'acs
(York University)

A commutator type operator in $l$ variables is a "multilinear" operator $T$ that maps any $l$-tuple of subgroups of an abelian group to a subgroup. Several typical examples for such operators are obtained from commutators in Lie-rings. We can define the concept of T-automorphism, $T$-solvability, $T$-nilpotency, $T$-ideal and $T$-section in a similar way as for Lie-rings.

My talk will focus on the following theorem: "If a finite $p$-group $P$ admits an automorphism of order $p^n$ with exactly $p^m$ fixed points, such that $\varphi^{p^{n-1}}$ has exactly $p^k$ fixed points, then $P$ has a fully-invariant subgroup of $(p,n,m,k)$-bounded index which is nilpotent of $m$-bounded class.". This is a theorem towards conjecture no. 2 of Medvedev's. After giving historical background and a brief explanation about what are these animals called "commutator type operator", I will explain how this theorem is obtained by applying twice a theorem on commutator type operators.

If the time will permit, I will show the idea of the proof of that theorem on commutator type operators.