# Difference between revisions of "Radical of a group"

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− | The largest [[Normal subgroup|normal subgroup]] of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771101.png" /> 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 | + | The largest [[Normal subgroup|normal subgroup]] of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771101.png" /> 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 | + | 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 [[#References|[1]]]). |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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) {{MR|0109842}} {{MR|0080089}} {{MR|0071422}} {{ZBL|0111.02502}} </TD></TR></table> |

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====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J.S. Robinson, "Finiteness conditions and generalized solvable groups" , '''1''' , Springer (1972) pp. 20ff {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.A.F. Wehrfritz, "Locally finite groups" , North-Holland (1973) pp. 12ff {{MR|0470081}} {{ZBL|0259.20001}} </TD></TR></table> |

In the class of Lie groups the radical is the largest connected solvable normal subgroup. In any Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771102.png" /> there is a radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771104.png" /> is a closed Lie subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771105.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771106.png" /> is a normal Lie subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771108.png" /> is semi-simple (see [[Lie group, semi-simple|Lie group, semi-simple]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771109.png" />. The subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711011.png" /> corresponding to the radical is the largest solvable ideal of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711012.png" />, called the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711013.png" />. | In the class of Lie groups the radical is the largest connected solvable normal subgroup. In any Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771102.png" /> there is a radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771104.png" /> is a closed Lie subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771105.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771106.png" /> is a normal Lie subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771108.png" /> is semi-simple (see [[Lie group, semi-simple|Lie group, semi-simple]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r0771109.png" />. The subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711011.png" /> corresponding to the radical is the largest solvable ideal of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711012.png" />, called the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077110/r07711013.png" />. | ||

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====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Satake, "Classification theory of semi-simple algebraic groups" , M. Dekker (1971) pp. 32ff {{MR|0316588}} {{ZBL|0226.20037}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.22001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table> |

## Revision as of 14:51, 24 March 2012

The largest normal subgroup of a group 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 [1]).

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) MR0109842 MR0080089 MR0071422 Zbl 0111.02502 |

#### Comments

#### References

[a1] | D.J.S. Robinson, "Finiteness conditions and generalized solvable groups" , 1 , Springer (1972) pp. 20ff |

[a2] | B.A.F. Wehrfritz, "Locally finite groups" , North-Holland (1973) pp. 12ff MR0470081 Zbl 0259.20001 |

In the class of Lie groups the radical is the largest connected solvable normal subgroup. In any Lie group there is a radical , and is a closed Lie subgroup in . If is a normal Lie subgroup in , then is semi-simple (see Lie group, semi-simple) if and only if . The subalgebra of the Lie algebra of corresponding to the radical is the largest solvable ideal of the Lie algebra , called the radical of .

The radical of an algebraic group , the largest connected solvable normal subgroup of , is always closed in . The radical of a linear algebraic group coincides with the connected component of the identity in the intersection of all Borel subgroups (cf. Borel subgroup) of ; it is the smallest closed normal subgroup such that is semi-simple (see Semi-simple algebraic group). The set of all unipotent elements in is a connected unipotent closed normal subgroup in , being the largest connected unipotent closed normal subgroup. This subgroup is called the unipotent radical of and can be characterized as the smallest closed normal subgroup in such that is reductive (cf. Reductive group).

*A.L. Onishchik*

#### Comments

#### References

[a1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff MR0251042 Zbl 0206.49801 Zbl 0186.33201 |

[a2] | I. Satake, "Classification theory of semi-simple algebraic groups" , M. Dekker (1971) pp. 32ff MR0316588 Zbl 0226.20037 |

[a3] | V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) MR0376938 Zbl 0371.22001 |

[a4] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039 |

**How to Cite This Entry:**

Radical of a group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Radical_of_a_group&oldid=11973