Difference between revisions of "Isolated subgroup"
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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 $M$. | 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 [[ | + | 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|Convex subgroup]]). | In the theory of ordered groups, isolated subgroups are sometimes referred to as convex subgroups (cf. [[Convex subgroup|Convex subgroup]]). | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.G. Kurosh, | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> A.G. Kurosh, "Theory of groups" , '''2''' , Chelsea (1960) pp. §66 (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 05:49, 20 June 2023
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) |
Isolated subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isolated_subgroup&oldid=32631