We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most $x$ are counted up to isomorphism and $x\to\infty$. More precisely, uniformly for $2^m\leq x<2^{m+1}$, all but an $O(2^{-cm^2})$ proportion have order $2^m$ and class two, for some absolute constant $c>0$. For each fixed prime $p$, we give a formula for the number of groups of order $p^m$, with relative error $O(p^{-m/3})$ as $m\to\infty$. This refines the classical logarithmic estimates for the number of $p$-groups. For every fixed prime $p$ and sufficiently large $m$, we also prove that all but a $p^{-m^2/300}$ proportion of groups of order $p^m$ have central elementary abelian Frattini subgroup. For fixed $p$, the distribution of the minimum number of generators among groups of order $p^m$ is asymptotically supported on one or two adjacent values, with explicit probabilities depending on $m$ modulo three.
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
In 2003, H\'ethelyi and K\"ulshammer proved that if $G$ is a finite solvable group and $p$ is a prime such that $p^2$ divides $|G|$, then $k(G)\geq (49p+1)/60$, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobeniu...
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
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...
We completely solve part (a) of Problem 17.112 in the Kourovka Notebook. We prove that a non-abelian finite simple group $G$ has a proper non-trivial ordinary $G$-permutable subgroup if and only if $G$ is isomorphic to ${\rm PSL_2(q)}$ where $q\geq4$ is a prime power and $q\not\equiv3\pmod4$, ${}^2B_2(2^{2a+1})$ where...
We extend Burnside's $p^a$-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size $2p^n$ for any prime $p$ and $n\geq 0$. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group $G$, the s...
Antonio Beltrán, Chang-Guo Shao· 0 citations
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