Unsolved Problems

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Product-Free Sets in Finite Groups

Which finite groups have the smallest largest product-free sets?...

Φ(G) and Φ'(G) Coincidence

Do $\Phi(G)$ and $\Phi'(G)$ coincide?...

Residually Finite Groups

Is every group well-approximated by finite groups?...

Large Subsets of Approximate Groups

Suppose $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$ with $|S| \gg K^{-O(1)}|A|$ and $S^8 \subset A^4$?...

Random Walk Mixing on Alternating Groups

Pick $x_1, \dots, x_k \in A_n$ at random. Is it true that, almost surely as $n \to \infty$, the random walk on this set of generators and their invers...

Bounds for Approximate Group Classification

Find bounds in the classification theorem for approximate groups....

Sofic Groups

Is every group well-approximated by finite groups?...