Order of an element in relation to the centre.

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Discussion Overview

The discussion revolves around the properties of elements in group theory, specifically focusing on whether an element of order 2 in a group must belong to the center of the group, Z(G). The scope includes theoretical aspects of group theory.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if an element of order 2 in a group must belong to the center Z(G).
  • Another participant asserts that it does belong to the center.
  • A hint is provided suggesting that there is a special element of order 2 in every group.
  • A counterpoint is raised that challenges the hint, clarifying that the identity element has order 1, not 2.
  • Participants express embarrassment over the misunderstanding regarding the identity element's order.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether an element of order 2 must belong to the center of the group, and there are competing views regarding the hints provided.

Contextual Notes

There are unresolved assumptions regarding the properties of elements in groups and the definitions of order and center.

sairalouise
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If a is the only element of order 2 in a group, does it belong to the centre Z(G)?
 
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Yes.
 
Hint: In every group, there is always a certain special (and familiar) element, of order 2.
 
cup said:
Hint: In every group, there is always a certain special (and familiar) element, of order 2.
Not true. If you're thinking of the identity element, then it has order 1.

I don't want to give the OP any hints because he/she hasn't posted any work.
 
morphism said:
Not true. If you're thinking of the identity element, then it has order 1.

You are right, of course. How embarrassing...
 
cup said:
You are right, of course. How embarrassing...
It happens to all of us! :wink:
 

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