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Gauge field theory with rank-one field $T_{\mu }$ is a quantum field theory that describes the interaction of elementary spin-1 particles, of which being massless to preserve gauge symmetry. In this paper, we give a generalized, extended study of abelian gauge field theory under successive rotor model in general $D$-dimensional flat spacetime for spin-1 particles in the context of higher-order derivatives. We establish a theorem that $n$ rotor contributes to the ${\square }^n{T}^{\mu }$ fields in the integration-by-parts formalism of the action. This corresponds to the transformation of gauge field $T^{\mu }\to {\square }^n{T}^{\mu }$ and gauge field strength $G_{\mu \nu }\to {\square }^n{G}_{\mu \nu }$ in the action. The $n=0$ case restores back to the standard abelian gauge field theory. The equation of motion and Noether’s conserved current of the theory are also studied.

We classify the abelian groups G, for which the following property holds: for every subgroup H, every ϕ ∈ Aut (H) has an extension ψ ∈ Aut(G). We also classify the infinite polycyclic-by-finite groups and the finite nilpotent 2′-groups having this property. Fuchs, Bertholf, Walls and Tomkinson did similar work for groups which have the property that homomorphisms of its subgroups extend to the whole group.

For modular abelian group algebras over a perfect field of prime characteristic, a connection is given between the direct factor problem and the isomorphism problem.

The isomorphism problem for modular abelian group algebras is solved for p-groups of countable p-length for p odd. For the prime p = 2, it is solved for all 2-groups.

A subgroup $H$ of a group $G$ is said to be an $IC\mathrm{\Phi }$-subgroup of $G$ if $H\cap [H,G]\le \mathrm{\Phi }(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\mathrm{\Phi }$-subgroups of $G$. A new characterization of finite abelian groups and some new criteria for $2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.

Let G be a finite group, we define the average codegree of the irreducible characters of G as $\mathrm{\text{acod}}(G)=\frac{1}{|\mathrm{\text{Irr}}(G)|}{\sum }_{\chi \in \mathrm{\text{Irr}}(G)}\mathrm{\text{cod}}(\chi )$, where $\mathrm{\text{cod}}(\chi )=\frac{|G:ker\chi |}{\chi (1)}$. We prove that if $G$ is non-solvable, then $\mathrm{\text{acod}}(G)\ge 68/5$, and the equality holds if and only if $G\cong A_5$. Also, we show that if $G$ is non-supersolvable, then $\mathrm{\text{acod}}(G)\ge 11/4$, and the equality holds if and only if $G\cong A_4$. In addition, we obtain that if $p$ is the smallest prime divisor of $\left|G\right|$, then $\mathrm{\text{acod}}(G)<p$ if and only if $G$ is an elementary abelian $p$-group.

We study the natural class of rings where each nilpotent element generates a nilpotent ideal, calling them the strongly 2-primal rings. We derive many basic properties of these rings, analyze their behavior under standard ring constructions and extensions, and taxonomize their relationship to other natural generalizations of commutativity. A slightly stronger condition is to assume that any nilpotent element generates a nilpotent ideal of the same index of nilpotence. We find that the difference between these two properties explains divergent behaviors in direct products, polynomial rings, Morita contexts, and other constructions.

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings $R$ are always “J-abelian” in the sense that each idempotent of $R$ maps onto a central idempotent in $R$/rad($R$). On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking $n$-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.