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| − | A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527901.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527902.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527903.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527905.png" />; in other words, if an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527906.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527907.png" />) is solvable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527908.png" />, then the solution lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i0527909.png" />. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279010.png" /> is said to be strongly isolated if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279011.png" /> the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279012.png" /> in the whole group lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279013.png" />. The isolator of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279014.png" /> of elements of a group is the smallest isolated subgroup containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279015.png" />. | + | {{TEX|done}} |
| | + | A subgroup $A$ of a group $G$ such that $g\in A$ whenever $g^n\in A$, $g^n\neq1$; in other words, if an equation $x^n=a$ (where $1\neq a\in A$) is solvable in $G$, then the solution lies in $A$. A subgroup $A$ is said to be strongly isolated if for every $a\in A$ the centralizer of $a$ in the whole group lies in $A$. The isolator of a set $M$ of elements of a group is the smallest isolated subgroup containing $M$. |
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| − | In an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279017.png" />-group (that is, in a [[Group with unique division|group with unique division]]), the concept of an isolated subgroup corresponds to that of a [[Pure subgroup|pure subgroup]] of an Abelian group. The intersection of isolated subgroups in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279018.png" />-group is an isolated subgroup. A normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279019.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279020.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279021.png" /> is isolated if and only if the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279022.png" /> is torsion-free. The centre of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052790/i05279023.png" />-group is isolated. | + | In an $R$-group (that is, in a [[Group with unique division|group with unique division]]), the concept of an isolated subgroup corresponds to that of a [[Pure subgroup|pure subgroup]] of an Abelian group. The intersection of isolated subgroups in an $R$-group is an isolated subgroup. A normal subgroup $H$ of an $R$-group $G$ is isolated if and only if the quotient group $G/H$ is torsion-free. The centre of an $R$-group is isolated. |
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| | In the theory of ordered groups, isolated subgroups are sometimes referred to as convex subgroups (cf. [[Convex subgroup|Convex subgroup]]). | | In the theory of ordered groups, isolated subgroups are sometimes referred to as convex subgroups (cf. [[Convex subgroup|Convex subgroup]]). |
Revision as of 08:28, 1 August 2014
A subgroup $A$ of a group $G$ such that $g\in A$ whenever $g^n\in A$, $g^n\neq1$; in other words, if an equation $x^n=a$ (where $1\neq a\in A$) is solvable in $G$, then the solution lies in $A$. A subgroup $A$ is said to be strongly isolated if for every $a\in A$ the centralizer of $a$ in the whole group lies in $A$. The isolator of a set $M$ of elements of a group is the smallest isolated subgroup containing $M$.
In an $R$-group (that is, in a group with unique division), the concept of an isolated subgroup corresponds to that of a pure subgroup of an Abelian group. The intersection of isolated subgroups in an $R$-group is an isolated subgroup. A normal subgroup $H$ of an $R$-group $G$ is isolated if and only if the quotient group $G/H$ is torsion-free. The centre of an $R$-group is isolated.
In the theory of ordered groups, isolated subgroups are sometimes referred to as convex subgroups (cf. Convex subgroup).
References
| [a1] | A.G. Kurosh, "Theory of groups" , 2 , Chelsea (1960) pp. §66 (Translated from Russian) |
How to Cite This Entry:
Isolated subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isolated_subgroup&oldid=32631
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article