- Equivalently, if and only if the
group equals its
abelianization. See
above for the
definition of a group's
abelianization. A
group G {\displaystyle G} is a perfect...
- so the
abelianization is trivial. For n < 3, An is trivial, and thus has
trivial abelianization. For A3 and A4 one can
compute the
abelianization directly...
-
homology group of a
group is
exactly the
abelianization of the group, and
perfect means trivial abelianization. An
advantage of this
definition is that...
- and a
subgroup H of
finite index, a
group homomorphism from G to the
abelianization of H. It can be used in
conjunction with the
Sylow theorems to obtain...
- in the
following section. Let G {\displaystyle G} be a p-group with
abelianization G / G ′ {\displaystyle G/G'} of
elementary abelian type ( p , p ) {\displaystyle...
-
index two
subgroups of the
hyperoctahedral group, as
discussed in H1:
Abelianization below, and
their intersection is the
derived subgroup, of
index 4 (quotient...
- The
first homology group is the
abelianization, and
corresponds to the sign map Sn → S2
which is the
abelianization for n ≥ 2; for n < 2 the symmetric...
- example). The
abelianization of a knot
group is
always isomorphic to the
infinite cyclic group Z; this
follows because the
abelianization agrees with the...
- {\text{Hom}}_{R}(M,-)} .
Given a
group G {\displaystyle G} , we can
define its
abelianization G ab = G / {\displaystyle G^{\text{ab}}=G/} [ G , G ] {\displaystyle...
- {\mathcal {G}}(p,1)}
exclusively contains p-groups G {\displaystyle G} with
abelianization G / G ′ {\displaystyle G/G^{\prime }} of type ( p , p ) {\displaystyle...
