Difference between revisions of "Radical of a group"
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+ | [[Normal subgroup|normal subgroup]] of a group $G$ belonging to a given radical class of groups. A class of groups is called radical if it is closed under homomorphic images and also under "infinite extension" , that is, if the class contains every group having an ascending normal series with factors from the given class (see | ||
+ | [[Normal series|Normal series]]). In every group there is a largest radical normal subgroup — the radical. The quotient group by the radical is a semi-simple group, that is, it has trivial radical. | ||
− | + | An example of a radical class is the class of groups having an ascending subnormal series with locally nilpotent factors. Sometimes the term "radical" is used just in connection with the largest locally nilpotent normal subgroup (in the case of finite groups this is the nilpotent radical or | |
− | + | [[Fitting subgroup|Fitting subgroup]]). The most important radical in finite groups is the solvable radical (see | |
− | + | [[Solvable group|Solvable group]]). Finite groups having a trivial solvable radical have a description in terms of simple groups and their automorphism groups (see | |
− | + | {{Cite|Ku}}). | |
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− | ==== | + | ====Radical of a Lie group==== |
+ | In the class of Lie groups the radical is the largest connected solvable normal subgroup. In any Lie group $G$ there is a radical $R$, and $R$ is a closed Lie subgroup in $G$. If $H$ is a normal Lie subgroup in $G$, then $G/H$ is semi-simple (see | ||
+ | [[Lie group, semi-simple|Lie group, semi-simple]]) if and only if $H\supset R$. The subalgebra of the Lie algebra $\def\fg{ {\rm \mathfrak g}}$ of $G$ corresponding to the radical is the largest solvable ideal of the Lie algebra $\fg$, called the radical of $\fg$. | ||
+ | ====Radical, unipotent radical of an algebraic group==== | ||
+ | The radical of an [[algebraic group]] $G$, the largest connected solvable normal subgroup of $G$, is always closed in $G$. The radical $R(G)$ of a linear algebraic group coincides with the [[connected component of the identity]] in the intersection of all [[Borel subgroup]]s of $G$; it is the smallest closed normal subgroup $H$ such that $G/H$ is [[Semi-simple algebraic group|semi-simple]]. The set $R(G)_{\rm u}$ of all unipotent elements in $R(G)$ is a connected [[Unipotent group|unipotent]] closed normal subgroup in $G$, being the largest connected unipotent closed normal subgroup. This subgroup is called the unipotent radical of $G$ and can be characterized as the smallest closed normal subgroup $H$ in $G$ such that $G/H$ is [[Reductive group|reductive]]. | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Linear algebraic groups", Benjamin (1969) pp. 283ff {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Linear algebraic groups", Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh, "The theory of groups", '''1–2''', Chelsea (1955–1956) (Translated from Russian) {{MR|0109842}} {{MR|0080089}} {{MR|0071422}} {{ZBL|0111.02502}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Sa}}||valign="top"| I. Satake, "Classification theory of semi-simple algebraic groups", M. Dekker (1971) pp. 32ff {{MR|0316588}} {{ZBL|0226.20037}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Va}}||valign="top"| V.S. Varadarajan, "Lie groups, Lie algebras, and their representations", Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.22001}} | ||
+ | |- | ||
+ | |} |
Latest revision as of 11:55, 23 November 2013
The largest
normal subgroup of a group $G$ belonging to a given radical class of groups. A class of groups is called radical if it is closed under homomorphic images and also under "infinite extension" , that is, if the class contains every group having an ascending normal series with factors from the given class (see
Normal series). In every group there is a largest radical normal subgroup — the radical. The quotient group by the radical is a semi-simple group, that is, it has trivial radical.
An example of a radical class is the class of groups having an ascending subnormal series with locally nilpotent factors. Sometimes the term "radical" is used just in connection with the largest locally nilpotent normal subgroup (in the case of finite groups this is the nilpotent radical or Fitting subgroup). The most important radical in finite groups is the solvable radical (see Solvable group). Finite groups having a trivial solvable radical have a description in terms of simple groups and their automorphism groups (see [Ku]).
Radical of a Lie group
In the class of Lie groups the radical is the largest connected solvable normal subgroup. In any Lie group $G$ there is a radical $R$, and $R$ is a closed Lie subgroup in $G$. If $H$ is a normal Lie subgroup in $G$, then $G/H$ is semi-simple (see Lie group, semi-simple) if and only if $H\supset R$. The subalgebra of the Lie algebra $\def\fg{ {\rm \mathfrak g}}$ of $G$ corresponding to the radical is the largest solvable ideal of the Lie algebra $\fg$, called the radical of $\fg$.
Radical, unipotent radical of an algebraic group
The radical of an algebraic group $G$, the largest connected solvable normal subgroup of $G$, is always closed in $G$. The radical $R(G)$ of a linear algebraic group coincides with the connected component of the identity in the intersection of all Borel subgroups of $G$; it is the smallest closed normal subgroup $H$ such that $G/H$ is semi-simple. The set $R(G)_{\rm u}$ of all unipotent elements in $R(G)$ is a connected unipotent closed normal subgroup in $G$, being the largest connected unipotent closed normal subgroup. This subgroup is called the unipotent radical of $G$ and can be characterized as the smallest closed normal subgroup $H$ in $G$ such that $G/H$ is reductive.
References
[Bo] | A. Borel, "Linear algebraic groups", Benjamin (1969) pp. 283ff MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
[Hu] | J.E. Humphreys, "Linear algebraic groups", Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039 |
[Ku] | A.G. Kurosh, "The theory of groups", 1–2, Chelsea (1955–1956) (Translated from Russian) MR0109842 MR0080089 MR0071422 Zbl 0111.02502 |
[Sa] | I. Satake, "Classification theory of semi-simple algebraic groups", M. Dekker (1971) pp. 32ff MR0316588 Zbl 0226.20037 |
[Va] | V.S. Varadarajan, "Lie groups, Lie algebras, and their representations", Prentice-Hall (1974) MR0376938 Zbl 0371.22001 |
Radical of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radical_of_a_group&oldid=11973