39 problems

Let $L$ be a finite nonabelian simple group, and let $G$ be a finite group. The groups $G$ and $L$ are isospectral when $\omega(G)=\omega(L)$, where $\omega(H)$ denotes the set of…

Let $S$ be a simple group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $S$. A group element is a $p$-element if its order is a power of $p$. Conder's conjecture. If…

Let $F_2$ be the free group on two generators. A group is fully residually finite simple if, for every finite subset of its nontrivial elements, there is a homomorphism to a finite…

Let $F_n$ denote the free group on $n$ generators. A subgroup $C\leq F_n$ is characteristic if it is preserved by every automorphism of $F_n$. The Baby Wiegold conjecture. For ever…

For a prime $p$, let $O_p(G)$ be the intersection of all Sylow $p$-subgroups of $G$. Say that $G$ has $(*)_p$ if there exist two Sylow $p$-subgroups whose intersection is contained…

Let $G$ be a finite group, and let $S$ be a non-abelian simple group. Write $\mu(G)$ for the set of proportions of $r$-singular elements of $G$ as $r$ ranges over the primes dividi…

Let $p$ be a prime and let $S$ be a finite non-abelian simple group whose order is divisible by $p$. Simple-group reduction conjecture. The following assertions hold. (a) Let $X$ b…

Let $G$ be a finite simple group and let $A\subseteq G$ be a subset with $|A|\geq 2$. A conjugate of $A$ means a set $gAg^{-1}$ for some $g\in G$. Liebeck–Nikolov–Shalev conjecture…

Almost-all non-congruence presentations conjecture. As the order of $S$ tends to infinity, the proportion of non-congruence presentations among all presentations tends to $1$:

Let $\Sigma$ be the infinite family of finite non-abelian simple groups … Suppose that a profinite group $G$ satisfies $P_2(G)>0$, where $P_2(G)$ denotes the proportion of 2-elemen…

Cocycle-Cheeger conjecture. The cocycle Cheeger constants in permutations of the Steinberg presentations have a uniform positive lower bound, at least when $n$ is fixed and…

Abdollahi–Akbari–Maimani conjecture. If

Vo-set characterization conjecture. $G\cong S$ if and only if

Two-orders characterization conjecture. $G\cong S$ if and only if

Thompson's conjecture. $G \cong S$ if and only if $|G|=|S|$ and $\pi_e(G)=\pi_e(S)$.

A graphical regular representation (GRR) of a group $G$ is a Cayley graph whose full automorphism group is isomorphic to $G$; a GRR is $k$-valent when the corresponding graph has v…

Let $c$ be a positive integer, and let $G$ be a finite non-abelian simple group. Write $\gamma(G)$ for its normal covering number. Bounded normal covering conjecture. There exists…

Let $G$ be a finite non-abelian simple group. Write $\gamma(G)$ for its normal covering number and $\gamma_w(G)$ for its weak normal covering number, and let $m\in\mathbb{N}$. Weak…

Fang–Xia conjecture. Except for a finite number of cases, every finite non-abelian simple group has a cubic GRR.

Let $G$ be a group and $S$ be a finite simple group. Let $pe(H)$ denote the invariant used in the source, namely the combined data of the relevant element-order counts. **NPE char…

Let $S$ be a finite simple group, and let $p$ be the largest prime divisor of $|S|$. For a group $H$, let $H(p)$ denote the set of elements of order $p$; equivalently, the number o…

Let $G$ be a non-abelian finite simple group, and let $S_1,\ldots,S_k$ be normal subsets of $G$, meaning unions of conjugacy classes. Gill–Pyber–Szabó's product conjecture. There e…

Let $G$ and $H$ be finite simple groups. For a finite group $K$, let $2\text{-}\operatorname{Cent}(K)$ denote the set of centralizers of $2$-element subsets of $K$, and let…

Interval-rank conjecture. The group $A_{11}$ is the only finite simple group whose set of ranks of string C-group representations is not an interval in the set of integers.

Let $X$ be a finite almost simple group, meaning that its socle $\operatorname{soc} X$ is a nonabelian simple group and…