80 problems

Does there exist a countable group that is not sofic?

Do a finite group’s order and the sum $\sum_{g\in G}\varphi(|g|)$ determine whether the group is simple?

Does there exist a group having more than one but only finitely many maximal locally soluble normal subgroups?

For an extension $G=A\rtimes B$ of elementary abelian $p$-groups with an element $a\in A$ satisfying $C_B(a)=1$, must $H=\langle a,B\rangle$ satisfy…

Can a nonabelian group admit a Rota–Baxter operator that is surjective but not injective?

Must the right-relatively convex subgroups of a right-orderable nonabelian group form a sublattice of its subgroup lattice?

Original wording: do Dlab groups have order automorphisms that are not inner? Strengthened wording: can such an automorphism fail to be induced by conjugation in any possibly large…

Does there exist a nonabelian simple group of Lie type that is totally $3$-closed?

For fixed distinct primes $p,q$, does every finite group embed into a finite group invariably generated by elements of orders $p$ and $q$?

Must every polynomial endomorphism of a rank-$n$ symplectic Poisson algebra that preserves the canonical bracket be an automorphism?

For which compact connected Lie groups is the kernel of Haar integration a Mathieu–Zhao space?

Do the leading residues of normalized Witten zeta functions for all irreducible crystallographic root systems admit the gamma-product form predicted by Au?

Is the depth of every finite-group cohomology ring realized by the dimension of one of its associated primes?

If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four?

If $\operatorname{CT}_{(k)}$ is generated by all horizontal class transpositions with modulus at most $k$, is $\operatorname{CT}_{(k)}\cong S_{\operatorname{lcm}(2,\ldots,k)}$ for…

Given positive integers $n$ and $1\leq k\leq n!$, can $n$ distinct group elements have exactly $k$ distinct values among all their permuted products?

If $\operatorname{Sol}_G(x)$ is a proper subgroup of a nonsolvable finite group $G$, must its intersection with its normalizer be metabelian?

If $\operatorname{Sol}_G(x)$ is a proper subgroup of a nonsolvable finite group $G$, must the intersection of all its conjugates lie in the hypercenter of $G$?

If $G$ is nonsolvable and $\operatorname{Sol}_G(x)\cap\operatorname{Sol}_G(y)$ is nonempty, must that intersection contain a nontrivial normal subgroup of $G$?

For the universal Schur coefficients $c_\lambda(n)$ in the multigraded Hilbert series of the coinvariant algebra $R_n^{(m)}$, give a combinatorial interpretation when $\lambda$ is…

For a pure O-sequence $h=(h_0,\ldots,h_e)$ of codimension three and type two, is $h_i^2\geq h_{i-1}h_{i+1}$ for every interior index $i$?

For which commutative rings is the property that all principal minors equal $1$ inherited by every integer power of a matrix?

If $\mathcal L(x^a y^b)=a!b!$ and $\mathcal L(f^m)=0$ for every positive integer $m$, must a polynomial $f\in\mathbb C[x,y]$ be zero?

Can one construct explicitly a complex Lie algebra isomorphic to its conjugate but not definable over $\mathbb R$, and determine the smallest two-step nilpotent dimension where thi…

In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field Fq of q elements. A polynomial f(x) in Fq[x] of degree d i…