commutator subgroup

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commutator subgroup

[′käm·yə‚tād·ər ′səb‚grüp]
(mathematics)
The subgroup of a given group G consisting of all products of the form g1 g2gn, where each gi is the commutator of some pair of elements in G.
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References in periodicals archive ?
If G is any group, let [G.sub.ab] denote the abelianization of G.
In [3], the authors provide a combinatorial method for calculating the abelianization of the discrete fundamental group of the permutahedron, which in turn gives a purely combinatorial method of calculating the Betti number of [M.sub.n,k].
When we move on to the abelianization of [A.sup.n-2.sub.1]([T.sub.n], [T.sub.0]), we are considering the equivalence classes of holes, corresponding to 5-cycles in [Asc.sub.n], but we are able to show that although there are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] equivalence classes, we may recover all of the equivalence classes of 5-cycles using only a set of (n+2) equivalence classes.
Theorem 1.1 The abelianization of [A.sup.n-2.sub.1]([T.sub.n], [T.sub.0]) is a free abelian group of rank [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
At each step in the truncation process, the number of generators of the abelianization of the discrete fundamental group increases, going from trivial in the case of the n-simplex to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the associahedron and [2.sup.n-3] ([n.sup.2] - 5n + 8) - 1 for the permutahedron.
Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math.
Discrete groups that are Kazhdan groups have finite abelianization [BdlHV08, Corollary 1.3.6] and finite groups are Kazhdan groups.
where [[pi].sup.ab.sub.1](X) stands for the abelianization [[pi].sub.1](X)/ [[[pi].sub.1](X), [[pi].sub.1](X)] of [[pi].sub.1](X).