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...

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...

characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center...

{\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...

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...

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...

gh = hg. commutator subgroup The commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. composition...

a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the...

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...

center of E and [ , ] denotes the commutator. Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group...

ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [am, bn]...

Commutator subgroup – Smallest normal subgroup by which the quotient is commutative Abelianization – Quotienting a group by its commutator subgroup Dihedral...

unique up to isomorphism. A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group F 2 {\displaystyle...

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...

and the commutator subgroup [ G , G ] {\displaystyle [G,G]} . More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal...

Lovász conjecture.) Cayley graphs on nilpotent groups with cyclic commutator subgroup are Hamiltonian. The flip graph of a convex polygon or equivalently...

L ( n , R ) {\displaystyle \mathrm {SL} (n,\mathbb {R} )} is the commutator subgroup of the general linear group G L ( n , R ) {\displaystyle \mathrm...

which is non-abelian, and the subgroup is an arithmetic subgroup and in particular does not contain the commutator subgroup. Commutativity of the convolution...

elements have only a single rotational/translational subgroup of order 2, which is also the commutator subgroup, examples [3,3]+, [3,5]+, [3,3,3]+, [3,3,5]+....

ring A {\displaystyle A} is the universal central extension of the commutator subgroup of the stable general linear group of A {\displaystyle A} . It is...

letters and denoted by An or Alt(n). For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements...

{\displaystyle G} is called powerful if the commutator subgroup [ G , G ] {\displaystyle [G,G]} is contained in the subgroup G p = ⟨ g p | g ∈ G ⟩ {\displaystyle...

Every metacyclic group is supersolvable. The commutator subgroup of a supersolvable group is nilpotent. Subgroups and quotient groups of supersolvable groups...

group homomorphisms as morphisms Metabelian group, a group where the commutator subgroup is abelian Abelianisation Abelian variety, a complex torus that can...

The special linear group is also the derived group (also known as commutator subgroup) of the GL(n, F) (for a field or a division ring F) provided that...

is the commutator subgroup of [4,3,3]. A high-index reflective subgroup is the prismatic octahedral symmetry, [4,3,2] (), order 96, subgroup index 4...

of a nilpotent non-abelian group. The center and the commutator subgroup of Q8 is the subgroup { e , e ¯ } {\displaystyle \{e,{\bar {e}}\}} . The inner...

manifold M, the group Diff(M, r) is perfect, i.e. equal to its own commutator subgroup, where Diff(M, r) is the group of C^r diffeomorphisms of a smooth...

of E/F is WE/F = WF/W c E  (where the superscript c denotes the commutator subgroup). For more details about Weil groups see (Artin & Tate 2009) or (Tate...

representations of G equals the number of conjugacy classes that G has. The commutator subgroup of G is the intersection of the kernels of the linear characters...