Abstract: The group isomorphism problem, i.e. the task of deciding whether two finite groups are isomorphic, remains one of the prominent and intriguing open problems from the area of computer algebra. In the talk I give a gentle introduction to the group isomorphism problem and its connections to graph theory. The isomorphism problem is one of the most fundamental problems in group theory for which we do not have efficient algorithmic tools. Despite decades of active research, its computational complexity is not well understood, even for very limited classes of groups.

In comparison to groups, combinatorial tools for the graph isomorphism problem are much more developed and they play a central role in our current understanding of the problem’s complexity. A major tool in this scope is the Weisfeiler-Leman algorithm, which is also explained in the talk. It provides a universal and effective combinatorial method for distinguishing non-isomorphic graphs with strong links to the structure theory of graphs. I present recent work in which we transfer the Weisfeiler-Leman algorithm to the setting of groups and highlight how the isomorphism invariants it computes relate to commonly used group invariants.