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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with [p.sub.ij], [q.sub.ij], [r.sub.ij] [member of] Z and M' is in the commutator subgroup of [[GAMMA].sub.g](2), which is in [[GAMMA].sub.g](4, 8), then
where [G.sub.1] = [G, G], [G.sub.i] = [G, [G.sub.i-1]], the ith commutator subgroup.
The subgroup of G generated by the set {[x, y]| x, : y [member of] G} is called the commutator subgroup of G and will be denoted G'.
Also, Pettet gave a more general statement proving that A(G) = 1 if Z(G) = 1 and the commutator subgroup [[gamma].sub.2] (G) = G (See [8]).
Then, K has a one-dimensional center, and hence the commutator subgroup [K.sup.s] := [K, K] is of codimension one in K.
Let [X.sup.ab] be the maximal abelian covering over X corresponding to the commutator subgroup [[[pi].sub.1](X), [[pi].sub.1](X)] by Galois theory.
For a group X, let X' be the commutator subgroup of X.