Centralizer-finding problem

Given data
Our universe is some group $$U$$ (such as a linear group or a permutation group) in which inverses can readily be computed.

A group $$G$$ in $$U$$ is specified by a generating set $$A$$. An element $$x$$ in $$U$$ is specified.

Goal
We need to determine the centralizer of $$x$$ in $$G$$.

Note that since we can in particular restrict $$x$$ to only an element of $$G$$, this really solves the problem of finding the centralizer of an element of the group if it is described using a faithful linear or permutation representation.

In this article, we discuss the centralizer-finding problem for cases where $$U$$ is the permutation group on a finite set $$S$$ of size $$n$$.

Equivalent group-finding problems

 * Group intersection problem: It turns out that the two problems are PTIME equivalent.


 * Set stabilizer problem: It turns out that the two problems are PTIME equivalent.


 * Partition stabilizer problem:

Equivalent decision problems

 * Conjugacy problem
 * Group factorization problem
 * Double coset equality problem

All the PTIME equivalences can be shown by using the fact that each is PTIME equivalent to the set stabilizer problem.