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Register a new userNumbers which are orders only of cyclic groups
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Authors
Paul Pollack
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We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $gcd(n,phi(n))=1$. With $C(x)$ denoting the count of cyclic $nle x$, ErdH{o}s proved that $$C(x) sim e^{-gamma} x/logloglog{x}, quadtext{as $xtoinfty$}.$$ We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincare, in descending powers of $logloglog{x}$, namely $$frac{e^{-gamma} x}{logloglog{x}} left(1-frac{gamma}{logloglog{x}} + frac{gamma^2 + frac{1}{12}pi^2}{(logloglog{x})^2} - frac{gamma^3 +frac{1}{4} gamma pi^2 + frac{2}{3}zeta(3)}{(logloglog{x})^3} + dots right). $$
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Refining a result of Erdos and Mays, we give asymptotic series expansions for the functions $A(x)-C(x)$, the count of $nleq x$ for which every group of order $n$ is abelian (but not all cyclic), and $N(x)-A(x)$, the count of $nleq x$ for which every group of order $n$ is nilpotent (but not all abelian).
For an element $g$ of a group $G$, an Engel sink is a subset $mathcal{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group.
The main local invariants of a (one variable) differential module over the complex numbers are given by means of a cyclic basis. In the $p$-adic setting the existence of a cyclic vector is often unknown. We investigate the existence of such a cyclic vector in a Banach algebra. We follow the explicit method of Katz, and we prove the existence of such a cyclic vector under the assumption that the matrix of the derivation is small enough in norm.
Let $G$ be a finite abelian group. We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called textit{purely singular} if for each prime divisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In this paper, we mainly study the purely singular splittings of cyclic groups. We first prove that if $kge3$ is a positive integer such that $[-k+1, ,k]^*$ splits a cyclic group $mathbb{Z}_m$, then $m=2k$. Next, we have the following general result. Suppose $M=[-k_1, ,k_2]^*$ splits $mathbb{Z}_{n(k_1+k_2)+1}$ with $1leq k_1< k_2$. If $ngeq 2$, then $k_1leq n-2$ and $k_2leq 2n-5$. Applying this result, we prove that if $M=[-k_1, ,k_2]^*$ splits $mathbb{Z}_m$ purely singularly, and either $(i)$ $gcd(s, ,m)=1$ for all $sin S$ or $(ii)$ $m=2^{alpha}p^{beta}$ or $2^{alpha}p_1p_2$ with $alphageq 0$, $betageq 1$ and $p$, $p_1$, $p_2$ odd primes, then $m=k_1+k_2+1$ or $k_1=0$ and $m=k_2+1$ or $2k_2+1$.
The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.
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