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We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power $q$ and every $m\geq1$, we construct a finite metabelian group $G_{m,q}$ with $Z_\infty(G_{m,q})=1$ and $|\Phi(G_{m,q})|=q$ for which the least number of system normalizers with trivial intersection is $m+1$. Thus the number...
<jats:p> In this paper, we prove that if <jats:italic>H</jats:italic> is a subgroup of a finite solvable group <jats:italic>G</jats:italic> , then <jats:italic>H</jats:italic>...
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.
We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic $2$, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component...
Albilich, an open-source agentic harness for autoresearch in mathematics that combines long-horizon reasoning, computer algebra systems (CAS), literature retrieval, and persistent SQLite-based context management, is presented.
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