Does Order Matter in Cyclic Subgroups/Groups?

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Discussion Overview

The discussion revolves around the question of whether the order of elements matters in cyclic subgroups or groups, particularly in the context of non-commutative operations. Participants explore the implications of the notation used to represent groups generated by elements.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • One participant questions whether the order of elements in a group matters, suggesting that may not equal due to non-commutative operations.
  • Another participant asserts that if refers to the group generated by a and b, then the order does not matter, as both and would represent the same group.
  • A clarification is made regarding the terminology, noting that 'generated' is the appropriate term for groups, while 'spanned' is typically reserved for vector spaces and modules.
  • A later reply expresses gratitude for the clarification and acknowledges the confusion regarding terminology.

Areas of Agreement / Disagreement

Participants express differing views on the implications of order in group notation, with some asserting that order does not matter in the context of generated groups, while others highlight the significance of non-commutativity in certain operations. The discussion remains unresolved regarding the broader implications of these concepts.

Contextual Notes

There are limitations in the discussion regarding the definitions of terms used, such as "generated" versus "spanned," and the implications of non-commutative operations in group theory. These aspects are not fully explored or resolved.

Gear300
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Since certain operations are not commutative, when a group G = <a, b>, does the order matter (so that <a, b> is not necessarily equal to <b, a>)?
 
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Assuming that by <a,b> you meant "the group spanned by a and b", the order does not matter. In that case the notation <a, b> -- as well as <b, a> -- means: take elements a and b and multiply them and their inverses until you get a group. Actually, it means: take the smallest group of which a and b are elements. In particular this means that both ab and ba must be elements of both <a, b> and <b, a>. Actually, a more concise notation would be: <{a, b}>
 
except in this case we say group 'generated' by a and b. for some reason 'spanned' is used only for vector spaces and (some times) modules.
 
I see...thanks for the replies.
 
Whoops, thanks nirax. Was already wondering why that sounded so odd ;)
 

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