Formula for Schur multiplier of free powered nilpotent group
Contents
Statement
Description as a lower central series quotient
Suppose is a set of primes.
Suppose is the free
-powered nilpotent group of class
on a generating set
. If
is the free
-group on
,
is explicitly the quotient group
, where
is the
member of the lower central series of
. Note that the numbering starts with
,
, and so on.
Then, the Schur multiplier of
is the group
. Note that this is an abelian
-powered group.
Explicit formula for dimension
Suppose is a free nilpotent group of class
on
generators. Then, the Schur multiplier
is a free
-module whose rank is given by the formula for dimension of graded component of free Lie algebra where we are talking of the
graded component for the free Lie algebra with
generators. The explicit formula is:
Related description of Schur covering group
The unique Schur covering group of the free -powered nilpotent group of class
on a set
is the free nilpotent group of class
on the same set
. Explicitly, if
is the free
-powered group on
and
is the free class
group on
, then:
-
- The Schur covering group of
is
The corresponding short exact sequence is:
Related description of exterior square
The exterior square of the free -powered nilpotent group of class
on a set
is defined as follows. Suppose
is the free group on
and
. is the free class
group on
, then:
Here, .
This group is not in general a free nilpotent group (note that the case is special). In general, the group is nilpotent of class
.
Particular cases
Description of Schur multiplier
The rows here correspond to the nilpotency class used and the columns correspond to the number of generators. The entry in the cell describes the Schur multiplier for the free nilpotent group of class given by the row and number of generators given by the column:
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0 | ![]() |
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0 | ![]() |
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0 | ![]() |
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0 | ![]() |
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0 | ![]() |
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Description of exterior square
Positive integer ![]() ![]() ![]() ![]() |
Hirsch length (equals sum of dimensions of graded components from the second to the ![]() |
Minimum size of generating set for the exterior square of the group | Nilpotency class of the exterior square (equals ![]() |
Comments |
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1 | ![]() |
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1 | we are starting with the free abelian group ![]() ![]() |
2 | ![]() |
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1 | we are starting with a free class two group but we end up with a free abelian group. |
3 | ![]() |
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2 | we are starting with the free class three group, but we end up with a non-free class two group. |
4 | ![]() |
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2 | we are starting with the free class four group, but we end up with a non-free class two group. |
Related facts
- Formula for nilpotent multiplier of free nilpotent group
- Formula for nilpotent multiplier of free powered nilpotent group
Facts used
Proof
Fact (1) says that if can be written as
where
is a free
-powered group, then we have:
In our case, . We thus get:
Since ,
. Further, by the definition of lower central series,
. Thus, we get: