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)...

Let $G$ be a $p$-mixed abelian group and $R$ is a commutative perfect integral domain of $charR=p>0$. Then, the first main result is that the group of all normalized invertible elements $V\left(RG\right)$ is a $\Sigma$-group if and only if $G$ is a $\Sigma$-group. In particular, the second central result is that if $G$ is a $\Sigma$-group, the $R$-algebras isomorphism $RA\cong RG$ between the group algebras $RA$ and $RG$ for an arbitrary but fixed group $A$ implies $A$ is a $p$-mixed abelian $\Sigma$-group and even more that the high subgroups of $A$ and $G$ are isomorphic, namely,...

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...

Let $G$ be a finite group and ${\pi }_e\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi }_e\left(G\right)$ and $m_k$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\{m_k:k\in {\pi }_e\left(G\right)\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups $L_2\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $\left|G\right|=|L_2\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, $p^2-1$, $p(p+ϵ)/2$ and some numbers divisible by $2p$, where $p$ is a prime...

Let G be an infinite abelian p-group and let K be a field of the first kind with respect to p of characteristic different from p such that $s_p\left(K\right)=\mathbb{N}$ or $s_p\left(K\right)=\mathbb{N}\cup 0$. The main result of the paper is the computation of the Ulm-Kaplansky functions of the factor group S(KG)/G of the normalized Sylow p-subgroup S(KG) in the group ring KG modulo G. We also characterize the basic subgroups of S(KG)/G by proving that they are isomorphic to S(KB)/B, where B is a basic subgroup of G.