Are Normal Subgroups Defined by Equality of Left and Right Cosets?

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Discussion Overview

The discussion revolves around the definition and properties of normal subgroups within group theory, specifically focusing on the relationship between left and right cosets. Participants explore the implications of subgroup order and the uniqueness of subgroups.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant states that a subgroup H is normal if gHg^(-1) = H for all g, suggesting a foundational property of normal subgroups.
  • Another participant proposes that if H is the unique subgroup of a certain order n, then it must be normal, as all conjugates xHx^(-1) would also have the same order n.
  • A different participant mentions that a subgroup H is normal if the left cosets and right cosets are equal, indicating a key characteristic of normal subgroups and its implications for defining group operations on cosets.

Areas of Agreement / Disagreement

Participants present multiple viewpoints regarding the definition and properties of normal subgroups, with no clear consensus reached on the implications of subgroup order or the relationship between cosets.

Contextual Notes

Some assumptions about subgroup orders and the uniqueness of subgroups are not fully explored, and the discussion does not resolve the implications of these conditions on the definition of normal subgroups.

raj123
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Normal subgroups??

Normal subgroups?
 
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H is normal if gHg^(-1)=H for all g. If H is a subgroup of some order, then so is gHg^(-1). End of hint.
 
Ah so

If H is unique subgroup of order n (no others) it must be normal as all other xHx^(-1) must be of that same order n.

I was thrown by the 10 or 20 in the problem, but it could really be any order n.

Thank you very much for the hint. I saw the disclaimer after I posted about the homework, so I'm sorry if this question wasn't up to par.
 
Also, a subgroup, H, of a group, G, is a normal subgroup if and only if the "left cosets" and "right cosets" are the same. A result of that is that we can define the group operation on the cosets (if p is in coset A and q is in coset B then AB is the coset that contains pq) in such away that the collection of cosets is a group in its own right: G/H.
 

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