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ehrenfest
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Homework Statement
Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true.
The commutator subgroup of a group G is the subgroup generated by all the commutators [a,b] where a and b are elements of G. It is denoted by G'.
The center of a group G is the set of elements that commute with every other element in G. It is denoted by Z(G) or C(G).
The commutator subgroup is a subgroup of the center. It is the smallest normal subgroup of G such that the quotient group G/G' is abelian. In other words, the commutator subgroup measures how "non-abelian" a group is, while the center measures how "central" a group is.
The commutator subgroup and center are important concepts in group theory as they provide information about the structure and properties of a group. They are used to classify groups into different types, such as abelian and non-abelian groups. They also play a crucial role in the study of group homomorphisms and group extensions.
Yes, the commutator subgroup and center can be equal for certain types of groups. For example, all abelian groups have trivial commutator subgroup and center, while some non-abelian groups such as the quaternion group have non-trivial commutator subgroups and centers that are equal.