Commutator subgroup

Algebraic Structures I (Matrices, Groups, Rings, and Fields)

Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019

Remark: [G,G] is called the commutator subgroup of G. In general, the commutators of a group G need not form a subgroup of G, and it is for this reason that we take the subgroup generated by the commutators of G. There is no known elementary counter-example. For a counter-example, see for instance, Theory of Groups [34].

∗-Algebras in Several Complex Variables

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Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019

Harald Upmeier

The Jordan algebra determinant N is invariant under the commutator subgroup K′. For any m ∈ N, the space Pm, …,m(Z) is 1-dimensional and consists of all multiples of Nm. Using the notation (5.30), one shows that Pr−1(Z)={ψ∈P(Z):∂Nψ=0}

Solvability, Supersolvability and Schreier Refinement Theorem for L-Subgroups

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Published in Fuzzy Information and Engineering, 2021

N. Ajmal, I. Jahan, B. Davvaz

Commutator subgroups and solvable groups are of considerable importance in classical algebra. In order to make the content of this work self sufficient, we recall some definitions and results in the context of these concepts from classical group theory which can be found in any standard text on the subject [15,16]. Let Then, the element is known as the commutator of a and b. If , then the commutator subgroup is defined by . The particular subgroup , generated by all the commutators in G, usually denoted by , is known as the derived subgroup (commutator subgroup) of G.

Commutator subgroup

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Algebraic Structures I (Matrices, Groups, Rings, and Fields)

∗-Algebras in Several Complex Variables

Solvability, Supersolvability and Schreier Refinement Theorem for L-Subgroups