Exterior square of finite group is finite: Difference between revisions
No edit summary |
No edit summary |
||
| Line 2: | Line 2: | ||
Suppose <math>G</math> is a [[finite group]]. Then, the [[fact about::exterior square of a group;2| ]][[exterior square]] of <math>G</math> is also a [[finite group]]. | Suppose <math>G</math> is a [[finite group]]. Then, the [[fact about::exterior square of a group;2| ]][[exterior square]] of <math>G</math> is also a [[finite group]]. | ||
==Applications== | |||
* [[Schur-Baer theorem]]: This states that if the [[inner automorphism group]] of a group is finite, so is the [[derived subgroup]] of the group. | |||
==Facts used== | ==Facts used== | ||
Latest revision as of 17:06, 29 June 2013
Statement
Suppose is a finite group. Then, the exterior square of is also a finite group.
Applications
- Schur-Baer theorem: This states that if the inner automorphism group of a group is finite, so is the derived subgroup of the group.
Facts used
- Commutator map is homomorphism from exterior square to derived subgroup, and the kernel of this homomorphism is the Schur multiplier.
- Schur multiplier of finite group is finite
Proof
The proof basically follows directly by combining Facts (1) and (2).