Suppose $R$ is a commutative ring with identity of prime characteristic $p$ and $G$ is an arbitrary abelian $p$-group. In the present paper, a basic subgroup and a lower basic subgroup of the $p$-component $U_p\left(RG\right)$ and of the factor-group $U_p\left(RG\right)/G$ of the unit group $U\left(RG\right)$ in the modular group algebra $RG$ are established, in the case when $R$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $p$-component $S\left(RG\right)$ and of the quotient group $S\left(RG\right)/G_p$ are given when $R$ is perfect and $G$ is arbitrary whose $G/G_p$ is $p$-divisible....

Let $S\left(RG\right)$ be a normed Sylow $p$-subgroup in a group ring $RG$ of an abelian group $G$ with $p$-component $G_p$ and a $p$-basic subgroup $B$ over a commutative unitary ring $R$ with prime characteristic $p$. The first central result is that $1+I(RG;B_p)+I(R\left(p^i\right)G;G)$ is basic in $S\left(RG\right)$ and $B[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]$ is $p$-basic in $V\left(RG\right)$, and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G_p/G_p$ is basic in $S\left(RG\right)/G_p$ and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G/G$ is $p$-basic in $V\left(RG\right)/G$, provided in both cases $G/G_p$ is $p$-divisible and $R$ is such that its maximal perfect subring $R^{p^i}$ has no nilpotents whenever $i$ is natural. The second major result is that $B(1+I(RG;B_p))$ is $p$-basic in $V\left(RG\right)$ and $(1+I(RG;B_p))G/G$ is $p$-basic in $V\left(RG\right)/G$,...

Suppose $F$ is a perfect field of $\mathrm{c}harF=p\ne 0$ and $G$ is an arbitrary abelian multiplicative group with a $p$-basic subgroup $B$ and $p$-component $G_p$. Let $FG$ be the group algebra with normed group of all units $V\left(FG\right)$ and its Sylow $p$-subgroup $S\left(FG\right)$, and let $I_p(FG;B)$ be the nilradical of the relative augmentation ideal $I(FG;B)$ of $FG$ with respect to $B$. The main results that motivate this article are that $1+I_p(FG;B)$ is basic in $S\left(FG\right)$, and $B(1+I_p(FG;B))$ is $p$-basic in $V\left(FG\right)$ provided $G$ is $p$-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...

In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal...

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM▹◂k\left(G\right)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{𝕆}$ as transversal in an order 128 group $X$ with subgroup ${\mathbb{Z}}_2^3$ and hence obtain a Hopf quasigroup $k{G}_{𝕆}>◂k\left({\mathbb{Z}}_2^3\right)$ as a particular case of our construction.

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...