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… g_n, where each g_i is the commutator of some pair of elements in G.

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perfect group

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).