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* For n \ge 3, SL_n(k) is a perfect group for any field k. * For n = 2, SL_n(k) is a perfect group if k has more than three elements. ... ...

The derived series or commutator series of a group is defined as follows: ... ordinal is the defining ingredient::commutator subgroup of its predecessor. ... ...

The perfect core or stable commutator of a group is all of the following ... # The limit of the (possibly transfinite) derived series. For a finite ... ...

perfect: Under the same conditions (n \ge 3 or k has at least four elements ... # uses::Perfectness is quotient-closed: The quotient of a perfect ... ...

a subgroup-defining function, namely the derived subgroup. ... * The trivial group is a perfect group. ===Groups satisfying the property=== ... ...

Category: Derived subgroups ==History== The notion of derived subgroup or commutator subgroup naturally arose ... ...

A group is said to be Schur-trivial or a group with trivial Schur ... # The homomorphism from its exterior square to its derived subgroup ... ...

if every coset of the defining ingredient::derived subgroup other than the ... * Weaker than::Perfect group * Weaker than::Abelian group ... ...

lower central series, Frattini series, derived series, Fitting series and ... the normal core of any Sylow subgroup or any Hall subgroup is characteristic. ... ...

! H in \! G by H \underline{\triangleleft} G or G \underline{\triangleright} H . In words, we say that \! H is normal in \! G or a normal subgroup ... ...