Is Using \leq for Subgroup Notation Incorrect?

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Discussion Overview

The discussion centers around the use of the notation \( H \leq S \) to denote that \( H \) is a subgroup of \( S \). Participants explore whether this notation is appropriate or incorrect, considering its implications and the conventions in mathematical literature.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the appropriateness of using \( \leq \) for subgroup notation, suggesting it may be misleading since the symbol typically represents a numerical relationship.
  • Another participant argues that it is acceptable to "overload" symbols in mathematics, provided that the usage is clearly defined.
  • Several participants note that the notation \( H \leq S \) is commonly used in various contexts, citing examples from online courses and personal experience.
  • Some participants mention that alternative notations, such as \( H \subset G \), are also used, but these may require additional context to clarify that \( H \) is a subgroup of \( G \).

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using \( \leq \) for subgroup notation. While some support its use and cite its prevalence, others raise concerns about potential confusion due to its traditional numerical meaning. The discussion remains unresolved regarding whether this notation is ultimately correct or incorrect.

Contextual Notes

There is a lack of consensus on the conventions surrounding subgroup notation, and participants highlight variations in usage across different texts and contexts. The discussion reflects the complexity of mathematical notation and its interpretation.

kntsy
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my lecturer use [itex]\leq[/itex] for subgroup.
For example
[itex]H \leq S[/itex] means H is a subgroup of S.
But is it a wrong use of notation as the less-than-equal sign is about number?
 
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No, there are, after all, only a finite number of symbols and an infinite number of possible concepts in mathematics! As long as you are careful to say how you are using a symbol, you can "overload" it.
 
Moreover, this is a frequent notation. See for instance this online course: http://user.math.uzh.ch/halbeisen/4students/gt.html"

Go to "Subgroups"
 
Last edited by a moderator:
arkajad said:
Moreover, this is a frequent notation. See for instance this online course: http://user.math.uzh.ch/halbeisen/4students/gt.html"

Go to "Subgroups"

In fact, I've never seen this notation not used.
 
Last edited by a moderator:
Newtime said:
In fact, I've never seen this notation not used.

In some books any special notation for H being a subgroup of G is carefully avoided. Words are always being used. In some other books it is written [tex]H\subset G[/tex] and you have to deduce from the context that H is a subgroup of G.
 

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