### A characterization of subgroup lattices of finite abelian groups.

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Suppose $F$ is a field of characteristic $p\ne 0$ and $H$ is a $p$-primary abelian $A$-group. It is shown that $H$ is a direct factor of the group of units of the group algebra $FH$.

We prove that pure subgroups of thick Abelian $p$-groups which are modulo countable are again thick. This generalizes a result due to Megibben (Michigan Math. J. 1966). Some related results are also established.

Suppose $p$ is a prime number and $R$ is a commutative ring with unity of characteristic 0 in which $p$ is not a unit. Assume that $G$ and $H$ are $p$-primary abelian groups such that the respective group algebras $RG$ and $RH$ are $R$-isomorphic. Under certain restrictions on the ideal structure of $R$, it is shown that $G$ and $H$ are isomorphic.

It is proved that if $G$ is a pure ${p}^{\omega +n}$-projective subgroup of the separable abelian $p$-group $A$ for $n\in N\cup \left\{0\right\}$ such that $|A/G|\le {\aleph}_{0}$, then $A$ is ${p}^{\omega +n}$-projective as well. This generalizes results due to Irwin-Snabb-Cutler (CommentṀathU̇nivṠtṖauli, 1986) and the author (Arch. Math. (Brno), 2005).

A new class of $p$-primary abelian groups that are Hausdorff in the $p$-adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, $V\left(G\right)/G$ is almost a coproduct of finite cyclic groups whenever $G$ is a Hausdorff $p$-primary group and $V\left(G\right)$ is the group of normalized units of the modular group algebra...

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...

A new class of abelian $p$-groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).