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A group G is said to be a B(n,k) group if for any n-element subset {a1,…, an} of G, |{aiaj | 1≤i, j≤n}|≤ k. In this paper, characterizations of B(4,11) groups, B(4,12) groups, and B(4,13) non-2-groups are given.

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

Let G be a group with cyclic derived subgroup of p-power order (p > 2), and F a field of characteristic p. We determine the derived length of the group of units of the group algebra FG.

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ8 has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

Let [Formula: see text] be a nontrivial [Formula: see text]-closed group and let [Formula: see text] be an arbitrary permutation group on a finite set [Formula: see text]. Let [Formula: see text] be the corresponding permutational wreath product of [Formula: see text] by [Formula: see text]. It is shown that every class-preserving Coleman automorphism of [Formula: see text]-power order of [Formula: see text] is inner. As a direct consequence, it is obtained that the normalizer property holds for [Formula: see text]. Further, it is shown that every class-preserving Coleman automorphism of [Formula: see text] is inner whenever [Formula: see text] is nilpotent. Our results generalize some known ones.

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

We consider a combinatorial problem in group theory. A group G is said to be a B(n, k) group if for any n-element subset A of G, ∣A2∣ ≤ k. In this paper, a complete characterization of B(4, 14) non-2-groups is given.

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.