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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453402.png" />-group'' | + | {{TEX|done}} |
| | + | ''$R$-group'' |
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| − | A group in which the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453403.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453404.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453405.png" /> are any elements in the group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453406.png" /> is any natural number. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453407.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453408.png" />-group if and only if it is torsion-free and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g0453409.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534010.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534011.png" /> and any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534012.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534013.png" />-group splits into the set-theoretic union of Abelian groups of rank 1 intersecting at the unit element. A group is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534014.png" />-group if and only if it is torsion-free and if its quotient group by the centre (cf. [[Centre of a group|Centre of a group]]) is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534015.png" />-group. Subgroups of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534016.png" />-group, as well as direct and complete direct products (cf. [[Direct product|Direct product]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534017.png" />-groups, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534018.png" />-groups. The following local theorem is valid for the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534019.png" />-groups: If all finitely-generated subgroups of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534020.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534021.png" />-groups, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534022.png" /> itself is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534023.png" />-group. Free groups, free solvable groups and torsion-free locally nilpotent groups (cf. [[Free group|Free group]]; [[Nilpotent group|Nilpotent group]]; [[Solvable group|Solvable group]]) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534024.png" />-groups. The class of all divisible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534025.png" />-groups (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045340/g04534027.png" />-groups, cf. also [[Divisible group|Divisible group]]) forms a variety of algebras under the operations of multiplication and division. | + | A group in which the equality $x^n=y^n$ implies $x=y$, where $x,y$ are any elements in the group and $n$ is any natural number. A group $G$ is an $R$-group if and only if it is torsion-free and is such that $x^ny=yx^n$ implies $xy=yx$ for any $x,y\in G$ and any natural number $n$. An $R$-group splits into the set-theoretic union of Abelian groups of rank 1 intersecting at the unit element. A group is an $R$-group if and only if it is torsion-free and if its quotient group by the centre (cf. [[Centre of a group|Centre of a group]]) is an $R$-group. Subgroups of an $R$-group, as well as direct and complete direct products (cf. [[Direct product|Direct product]]) of $R$-groups, are $R$-groups. The following local theorem is valid for the class of $R$-groups: If all finitely-generated subgroups of a group $G$ are $R$-groups, then $G$ itself is an $R$-group. Free groups, free solvable groups and torsion-free locally nilpotent groups (cf. [[Free group|Free group]]; [[Nilpotent group|Nilpotent group]]; [[Solvable group|Solvable group]]) are $R$-groups. The class of all divisible $R$-groups ($D$-groups, cf. also [[Divisible group|Divisible group]]) forms a variety of algebras under the operations of multiplication and division. |
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| | ====References==== | | ====References==== |
| | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR></table> | | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR></table> |
Latest revision as of 11:47, 29 June 2014
$R$-group
A group in which the equality $x^n=y^n$ implies $x=y$, where $x,y$ are any elements in the group and $n$ is any natural number. A group $G$ is an $R$-group if and only if it is torsion-free and is such that $x^ny=yx^n$ implies $xy=yx$ for any $x,y\in G$ and any natural number $n$. An $R$-group splits into the set-theoretic union of Abelian groups of rank 1 intersecting at the unit element. A group is an $R$-group if and only if it is torsion-free and if its quotient group by the centre (cf. Centre of a group) is an $R$-group. Subgroups of an $R$-group, as well as direct and complete direct products (cf. Direct product) of $R$-groups, are $R$-groups. The following local theorem is valid for the class of $R$-groups: If all finitely-generated subgroups of a group $G$ are $R$-groups, then $G$ itself is an $R$-group. Free groups, free solvable groups and torsion-free locally nilpotent groups (cf. Free group; Nilpotent group; Solvable group) are $R$-groups. The class of all divisible $R$-groups ($D$-groups, cf. also Divisible group) forms a variety of algebras under the operations of multiplication and division.
References
| [1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
How to Cite This Entry:
Group with unique division. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_with_unique_division&oldid=32342
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article