We show that any block of a group algebra of some finite group which is of wild representation type has many families of stable tubes.

Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G=G_p\times B$ is a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $S^{\lambda }G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $S^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $S^{\lambda }{G}_p$-module V and an irreducible $S^{\lambda }B$-module...

We characterize the unit group of semisimple group algebras ${𝔽}_{q}G$ of some non-metabelian groups, where $F_q$ is a field with $q=p^k$ elements for $p$ prime and a positive integer $k$. In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_3\times C_3)⋊C_3)⋊C_2$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...

Let $G$ be a finite group and $k$ a field of characteristic $p>0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_0$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_0$ has the property that $e_0{\left(k_P\right)}^G$ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_P$ is the trivial $kP$-module, ${\left(k_P\right)}^G$ is the induced module, and $e_0$ is the block idempotent of $B_0$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more than three...

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B 4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B 4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation...

In this paper we study restricted Boolean rings and group rings. A ring $R$ is $\mathrm{𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑𝐵𝑜𝑜𝑙𝑒𝑎𝑛}$ if every proper homomorphic image of $R$ is boolean. Our main aim is to characterize restricted Boolean group rings. A complete characterization of non-prime restricted Boolean group rings has been obtained. Also in case of prime group rings necessary conditions have been obtained for a group ring to be restricted Boolean. A counterexample is given to show that these conditions are not sufficient.

A G-shift of finite type (G-SFT) is a shift of finite type which commutes with the continuous action of a finite group G. For irreducible G-SFTs we classify right closing almost conjugacy, answering a question of Bill Parry.

In this note we obtain a necessary and sufficient condition for a ring to be $s$-weakly regular (i) When $R$ is a ring with identity and without divisors of zero (ii) When $R$ is a ring without divisors of zero. Further it is proved in a $s$-weakly regular ring with identity and without units every element is a zero divisor.