# Sysak Ya. P.

### On local near-rings with Miller?Moreno multiplicative group

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 811-818

A near-ring $R$ with identity is local if the set $L$ of all its noninvertible elements is a subgroup of the additive group $R^{+}$. We study the local near-rings of order $2^n$ whose multiplicative group $R^{*}$ is a Miller-Moreno group, i.e., a non-abelian group all proper subgroups of which are abelian. In particular, it is proved that if $L$ is a subgroup of index $2^m$ in $R^{+}$, then either $m$ is a prime for which $2^m - 1$ is a Mersenna prime or $m = 1$. In the first case $n = 2m$, the subgroup $L$ is elementary abelian, the exponent of $R^{+}$ does not exceed 4, and $R^{*}$ is of order $2^m(2^m - 1)$. In the second case either $n < 7$ or the subgroup $L$ is abelian and $R^{*}$ is a nonmetacyclic group of order $2^{n−1}$ and of exponent at most $2^{n−4}$.

### Radical algebras subgroups of whose adjoint groups are subalgebras

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1646–1652

We obtain the characteristic for radical algebras subgroups of whose adjoint groups are subalgebras. In particular, we prove that the algebras of this sort are nilpotent with nilpotent length at most three. We give the complete classification of those algebras under consideration which are generated by two elements.

### On ascending and subnormal subgroups of infinite factorized groups

De Glovanni F., Franclosi S., Sysak Ya. P.

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 842–848

We consider an almost hyper-Abellan group *G* of a finite Abelian sectional rank that is the product of two subgroups *A* and *B*. We prove that every subgroup *H* that belongs to the intersection *A* ∩ *B* and is ascending both in *A* and *B* is also an ascending subgroup in the group *G*. We also show that, in the general case, this statement is not true.