Let $r>1$ with $\gcd(r,30)=1$. We construct a finite group $G_r=(C_5\times C_3\times C_r)\rtimes W$, where $|W|=32$ and $G_r'\cong C_{60r}$, together with an augmentation-preserving automorphism $\alpha_r\in\operatorname{Aut}(\mathbb{Z}G_r)$ having no Zassenhaus factorization. The image $Y_r=\alpha_r(G_r)$ is a normalized group basis which is not rationally conjugate to $G_r$, although every element of $Y_r$ is individually rationally conjugate to an element of $G_r$. Consequently, (ZC2) and (ZC3) fail for finite cyclic-by-abelian groups, resolving a problem of Margolis and del R\'io. The construction extends Hertweck's example uniformly: both the class-preserving obstruction and the integral gluing are independent of the order of the auxiliary factor. The smallest admissible member of this family has order $3360$ and derived subgroup $C_{420}$.
Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n...
Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p...
Let $p_1,\dots,p_r\geq 5$ be distinct primes, let $P=p_1\cdots p_r$, and put $M=6P$. For a positive integer $n$, let $g_P(2n)$ denote the number of unordered representations $2n=h+k$, with $1\leq h\leq k$, such that $\gcd(h,M)=\gcd(k,M)=1$. Using the canonical remainder operator $\delta_q(x)=x-q\lfloor x/q\rfloor$, we...
The shifted congruence series of finite special Nottingham jet groups is realized by an integral augmentation filtration. For volume-preserving formal substitutions in characteristic $p\ge5$ and $n\ge3$ variables, the filtration induced by the action on truncated power series has the congruence subgroups as its dimensi...
Let $G=\Cay(\mathbb{Z}_n,S)$ be a $3$-valent circulant digraph with connection set $S=\{0,s,t\}$, where $s,t\in\mathbb{Z}_n^*$ and $t\neq\pm s$. Determining the automorphism group of its endomorphism monoid reduces to determining the subgroup $U_S(\mathbb{Z}_n)\leq\mathbb{Z}_n^*$ that normalizes $\End(G)$. In this pape...
Let $A$ be an abelian group and $E=\Endo_{\Z}(A)$. We give a module-theoretic description of the singular $E$-submodules asked for in Fuchs'Problem~1.2. For a unital ring $R$ and a left $R$-module $M$, put $W=I_R({}_RR)\oplus I_R(M)$ and $J=\Jaco(\Endo_R(W))$, and let $u$ be the image of $1_R$. The classical essential-...
Jiang Yang, Xin Zhang· 0 citations
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