With help of group characters we enumerate orientably-regular maps of a given type with automorphism group isomorphic to $\PGam(2,2^p)$ for prime $p$.
We obtain a characterization of finite non-$p$-groups with five automorphism orbits. In the solvable case of Fitting height $3$, the groups are precisely $S_4$ and $C_2^4\rtimes D_{10}$. Moreover, the corresponding fixed-point subspaces form a spread in a finite projective space. For Fitting height $2$, we distinguish...
We prove that a Hall $\pi$-subgroup $H$ of a finite group $G$ is normal if and only if every irreducible constituent of the permutation character $1_H^G$ has $\pi'$-degree.
Let $C$ be an $[n,k]_q$ maximum distance separable (MDS) code with $4\le k\le q-3$, and suppose that it has a $2$-transitive permutation automorphism group. In this paper we show that $n\le q+1$, so the MDS conjecture holds for this class of codes.
For any infinite cardinal $\kappa$, we give an example of a centerless group $G$ of cardinality $\kappa$ whose automorphism tower terminates after $\kappa^+$ steps, answering a question of Simon Thomas. The work on this paper was assisted by AI.
We prove that $\mathrm{Mod}(S_g)$ acts faithfully by orientation-preserving homeomorphisms on a $(2g-1)$-sphere when $g \geq 3$ using equivariant blow-up and bi-orderability of the Torelli group.
For $n$ odd, we enumerate rank $n$ complex topological vector bundles on $\mathbb CP^{n+2}$ with prescribed complex $K$-theory class. The answer exhibits a 24-fold periodicity and is closely related to the enumeration of corank $2$ stably trivial vector bundles.
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