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 g2… gn, where each gi is the commutator of some pair of elements in G.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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References in periodicals archive ?
If G is any group, let [G.sub.ab] denote the abelianization of G.
On the residual solvability of generalized free products of solvable groups
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.
The discrete fundamental group of the associahedron
Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math.
On some Hasse principles for algebraic groups over global fields
Discrete groups that are Kazhdan groups have finite abelianization [BdlHV08, Corollary 1.3.6] and finite groups are Kazhdan groups.
On Kazhdan's property (T) for the special linear group of holomorphic functions
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).
Nilpotent class field theory for manifolds

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