the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...
11 KB (1,833 words) - 17:10, 24 April 2023
commutators is closed and is called the derived group or the commutator subgroup of G. Commutators are used to define nilpotent and solvable groups and the...
14 KB (2,555 words) - 23:05, 8 August 2024
characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center...
10 KB (1,191 words) - 16:31, 29 June 2022
group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients...
10 KB (1,364 words) - 19:38, 22 May 2024
Solvable group (redirect from Solvable subgroup)
G^{(2)}\triangleright \cdots ,} where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of G. These two definitions...
18 KB (3,073 words) - 21:40, 17 August 2024
Transfer (group theory) (section Commutator subgroup)
{\displaystyle \textstyle \prod _{i=1}^{n}h_{i}} in H/H′, where H′ is the commutator subgroup of H. The order of the factors is irrelevant since H/H′ is abelian...
5 KB (786 words) - 03:58, 13 July 2023
Special linear group (section Lie subgroup)
related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and...
11 KB (1,481 words) - 01:34, 27 July 2024
and the commutator subgroup [ G , G ] {\displaystyle [G,G]} . More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal...
19 KB (3,157 words) - 18:43, 9 May 2024
Abelian group (redirect from Abelian subgroup)
Commutator subgroup – Smallest normal subgroup by which the quotient is commutative Abelianization – Quotienting a group by its commutator subgroup Dihedral...
36 KB (5,284 words) - 20:43, 1 August 2024
Central series (category Subgroup series)
central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence...
14 KB (2,194 words) - 02:35, 29 June 2024