Let $d(G)$ denote the least size of a generating set of a finite group $G$. We prove that if $G$ has a family $\mathcal H$ of subgroups such that $d(H)\leq d$ for every $H\in\mathcal H$ and $\gcd\{\lvert G:H\rvert:H\in\mathcal H\}=1$, then $d(G)\leq d+1$. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow $2$-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow $2$-subgroups of finite simple groups.
For a nontrivial finite abelian group $G$, let $\nu(G)$ be the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. It is...
A. Geroldinger, Guo-Qing Wang, Wenkai Yang· 1 citation· ⚡1
Let $G$ be a finite solvable group such that, for any prime $p$ and any quotient $Q$ of $G$, there are two Sylow $p$-subgroups of $Q$ intersecting in $O_p(Q)$. Then, for every family $(P_i)_{i=1}^n$ of Sylow subgroups of $G$ for distinct primes $p_1,\ldots,p_n$, there exists $x \in G$ such that $P_i \cap P_i^x = O_{p_i...
Hong-Yi Huang, Francesca Lisi, A. Rizzoli et al.· 0 citations
Ozaki proved that every finite $p$-group occurs as the Galois group of a maximal unramified $p$-extension of a number field. Hajir, Maire and Ramakrishna made this theorem effective, obtaining a base-field degree of order $|G|$. In this article, we reduce that degree by taking the Frattini structure of $G$ into account...
We show that for every finite group $G$ and every integer $n$, the Nil group $\mathrm{NK}_n(\mathbb{Z}[G])$ of the integral group ring of $G$ has finite exponent, and give a bound depending on n and the order $|G|$. This bound is explicit when every prime divisor of $|G|$ is at least $3+n/2$. More generally, our finite...
T. Chinburg, M. Morrow, G. Pappas et al.· 0 citations
For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ act...
A group is called an $n$-$\bfc$-group if each of its conjugacy classes is finite and contains at most $n$ elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of $n$-bounded order, whereas the order of the group itself need not be bounded, as extraspecial $p$-groups show. We prove that...
I. Gorshkov· 0 citations
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