Commutator subgroup and center

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Homework Help Overview

The discussion revolves around the relationship between the center of a group and its commutator subgroup, specifically questioning whether the center always contains the commutator subgroup. Participants are examining group theory concepts.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • One participant asserts the claim that the center contains the commutator subgroup, while another counters this by suggesting that such a claim would imply commutators commute, which is not necessarily true. A third participant references a theorem related to normal subgroups and abelian groups, raising questions about the implications of this theorem. Another participant provides a counterexample involving the alternating group A_5 to illustrate that G/Z(G) is not necessarily abelian.

Discussion Status

The discussion is active, with participants presenting differing viewpoints and counterexamples. There is no clear consensus, but several lines of reasoning are being explored regarding the relationship between the center and the commutator subgroup.

Contextual Notes

Participants are working under the constraints of group theory definitions and theorems, with some assumptions about normal subgroups and their properties being questioned.

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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.


Homework Equations





The Attempt at a Solution

 
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It's not. That would imply commutators commute, which there's no good reason to expect, and it should be easy to find a counterexample.
 
But my book has a theorem that says:

"If N is a normal subgroup of G, then G/N is abelian if and only if C is contained in N"
where C is the commutator subgroup.

Clearly the center is normal and its quotient group is abelian!
 
G/Z(G) is not necessarily abelian, e.g. Z(A_5) is trivial (since A_5 is simple), whence A_5/Z(A_5) =~ A_5, a non-abelian group.
 

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