Group satisfying Tits alternative: Difference between revisions
No edit summary |
|||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
== | ==Definition== | ||
A group is said to satisfy the '''Tits alternative''' if every subgroup of it is | A group is said to satisfy the '''Tits alternative''' if for every subgroup of it, one of these two conditions holds: | ||
# The subgroup is [[fact about::virtually solvable group|virtually solvable]] (i.e., has a [[solvable group|solvable]] [[subgroup of finite index]]) | |||
# The subgroup [[fact about::group having a free non-abelian subgroup|contains a free non-abelian subgroup]] (which is equivalent to saying that it contains a copy of [[free group:F2]]). | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 06:07, 7 June 2012
Definition
A group is said to satisfy the Tits alternative if for every subgroup of it, one of these two conditions holds:
- The subgroup is virtually solvable (i.e., has a solvable subgroup of finite index)
- The subgroup contains a free non-abelian subgroup (which is equivalent to saying that it contains a copy of free group:F2).
Relation with other properties
Stronger properties
- Finite group
- Abelian group
- Nilpotent group
- Solvable group
- Virtually Abelian group
- Virtually nilpotent group
- Virtually solvable group
- Free group