Open-problem statements, with their sources attached.

The zero-divisor conjecture*

If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.

If $G$ is a group, can there exist an exact covering of $G$ by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.)

The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.

The idempotent conjecture*

If G is torsion-free, then K[G] has no non-trivial idempotents.

Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left cosets of subgroups $G_1, \dots, G_k$ of $G$.

Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$, then the indices $[G:G_1], \dots, [G:G_k]$ cannot be distinct.

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

What is the largest product-free set in the alternating group $A_n$?

Let $G$ and $H$ be finite groups of the same order with $\sum_{g \in G} \phi(|g|) = \sum_{h \in H} \phi(|h|)$, where $\phi$ is the Euler totient function. Suppose that $G$ is simple. Is $H$ necessarily simple?

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most $p^{2k}$?

Markel's $S_3$-conjecture* (1973): any nontrivial finite ah-group is isomorphic to $S_3$.

The conjecture is open in general; it is known to be true for solvable groups.

Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$, $g^n = 1$. Is $G$ necessarily finite?

If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least $e^{\sqrt n}$ in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.

Conjecture:* Are there infinitely many Leinster groups?

This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups.

Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".

Gottschalk's surjunctivity conjecture* (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.

Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?

Note: A is taken as $\alpha$-dense, i.e. $|A| \ge \alpha |G|^2$ [Au16, Question 2]

Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$, together with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \ge n^{2-o(1)}$ and $|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$ ($j \neq k$)?

NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for $i \neq k$. See green_36.variants.cks05.

Variant using the exact simultaneous double product property from [CKS05, 4.1].

Which finite groups have the smallest biggest product-free sets?

We formalise this as: determine the supremum of exponents $\alpha$ such that every nontrivial finite group of order $n$ contains a product-free set of size $\geq c n^{\alpha}$ for some absolute constant $c > 0$. (The trivial group is excluded since its only product-free subset is empty.) Kedlaya [Ke97] showed that $\alpha = 11/14$ is admissible, and Green suggests this exponent may well be sharp; the candidate extremal family is the Ree groups ${}^2G_2(q)$, $q = 3^{2m+1}$.

A good model problem would be to determine the largest product-free subsets of $\mathrm{SL}_2(\mathbb{F}_p)$.

Babai–Seress Conjecture (Conjecture 1.5)**: There exists an absolute constant $C$ such that the diameter of the alternating group $A_n$ satisfies $\operatorname{diam}(A_n) \leq n^C.$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.5

Babai–Seress Conjecture (Conjecture 1.7)**: There exists an absolute constant $C$ such that every finite simple non-abelian group $G$ satisfies $\operatorname{diam}(G) \leq (\log |G|)^C.$ Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.7