Definitions

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This page collects terms used the Kourovka Notebook which might be unfamiliar or quaint to English-speaking mathematicians. Some terms may appear both in the Notebook and is used in modern English, but might carry a slightly different meaning. For example, in the Notebook, a divisible group is equal to a radicable group, which is not necessarily abelian. While in modern English usage, a divisible group is assumed to be abelian.

We keep the problem statements and remarks from the Notebook as they are, not changing them to more familiar terms.

Divisible, radicable groups. The Notebook seems to use these adjectives interchangeably. Indeed, they describe the same property for a group $G$: that the equation $x^n = g$ admits a solution for all integers $n\ge 1$ and all $g\in G$. However, in modern English usage, a divisible group predominantbly mean an abelian group with the said property, while the adjective radicable is for general (not necessarily abelian) groups.

Skew field = division ring.

Locally normal group, a group such that every finitely generated subgroup is contained in a finite normal subgroup.

One can give a more universal treatment of the word "locally". Let P be a group property. We say a group $G$ is locally P if every finitely generated subgroup of $G$ is contained in a subgroup with property P. If P hereditary to subgroup, this is the same as requiring that every finitely generated subgroup of $G$ has property P. This is precisely the definition for locally finite group, locally solvable group, locally nilpotent group, locally cyclic group, locally free group, locally polycyclic group. This characterization also fits the definition of locally simple group, locally symmetric group. Caveat: this characterization fails for "locally normal". Indeed, locally normal (Russian-school) = locally finite-normal (in our characterization).

Regular automorphism = fixed-point-free automorphism, that is, an automorphism whose set of fixed points is $\{1\}$.

Problem of the isomorphism to the trivial group. This is the problem of determining a group given by a presentation is trivial.

Rank of a group $G$ = special rank = Mal'cev rank = Prüfer rank(?). This is the smallest integer $r$ (if exists) such that every finitely generated subgroup of $G$ can be generated by at most $r$ elements.

Almost P. Let P be a group property. In the Russian school, a group $G$ called almost P if it admits a subgroup of finite index having property P. In modern English, it is called virtually P. For example, an group is almost abelian (virtually abelian) if it contains an abelian subgroup of finite index.

Maximum condition for/on subgroups = Noetherian property. That is, the ascending chain condition for subgroups. Equivalently, every subgroup is finitely generated.

Minimum condition for/on subgroups = Artinian property. That is, the descending chain condition for subgroups.

Quasi-invariant Subgroup = quasinormal subgroup = permutable subgroup. If subgroup $H$ of a group $G$ is quasi-invariant if $HK = KH$ for all subgroups $K$ of $G$.

N-group, a group satisfying the normalizer condition.

Binary nilpotent group, a group such that every $2$-generated subgroup is nilpotent.

R-group, a group with the unique root property. We say a group $G$ has the unique root property if $x^n = y^n$ for some $n\ge 1$ implies $x=y$.

Nilelement = Engel element, Nilgroup = Engel group.

Nil-automorphism. An automorphism $\varphi$ of a group $G$ is called a nil-automorphism if for any $g\in G$, there is an integer $n=n(g,\varphi)$ such that $[g, {}_n\varphi]=1$ in $G\rtimes \text{Aut}(G)$, the holomorph of $G$. That is, $\varphi$ acts as a left-Engel automorphism.