properties of group commutators and commutator subgroups
The purpose of this entry is to collect properties of http://planetmath.org/node/2812group commutators and commutator subgroups. Feel free to add more theorems!
Let be a group.
Let , then .
Direct computation yields
Let be subsets of , then .
By Theorem 1, the elements from or are products of commutators of the form or with and . ∎
Theorem 3 (Hall–Witt identity).
Let , then
This is mainly a brute-force calculation. We can easily calculate the first factor explicitly using theorem 1:
Let , the “first half” of . Let be the element obtained from by the cyclic shift , and be the element obtained from by . We have
which gives us
and, by applying twice
In total, we have
Theorem 4 (Three subgroup lemma).
The group is generated by all elements of the form with , and . Since is normal, and are elements of . The Hall–Witt identity then implies that is an element of as well. Again, since is normal, which concludes the proof. ∎
For any we have
The other identities are proved similarly. ∎
|Title||properties of group commutators and commutator subgroups|
|Date of creation||2013-03-22 15:30:50|
|Last modified on||2013-03-22 15:30:50|
|Last modified by||GrafZahl (9234)|
|Defines||three subgroup lemma|