Does Order Matter in Cyclic Subgroups/Groups?

  • Thread starter Gear300
  • Start date
  • Tags
    Cyclic
In summary, the order of elements in a group G = <a, b> does not matter. Both <a, b> and <b, a> mean to take the smallest group containing elements a and b, where multiplication of elements and their inverses results in a group. The notation <{a, b}> can also be used to represent this, but "generated" is typically used instead of "spanned" for groups.
  • #1
Gear300
1,213
9
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>)?
 
Mathematics news on Phys.org
  • #2
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}>
 
  • #3
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.
 
  • #4
I see...thanks for the replies.
 
  • #5
Whoops, thanks nirax. Was already wondering why that sounded so odd ;)
 

What is a cyclic subgroup?

A cyclic subgroup is a subgroup of a group that is generated by a single element. This means that all elements in the subgroup can be obtained by repeatedly applying the group operation to the generator element.

How is a cyclic subgroup different from a regular subgroup?

A regular subgroup can be generated by multiple elements, while a cyclic subgroup can only be generated by a single element. Additionally, all cyclic subgroups are abelian (commutative), while regular subgroups may not be.

Can a cyclic subgroup be infinite?

Yes, a cyclic subgroup can be infinite. This happens when the generator element has infinite order, meaning that it can be multiplied with itself an infinite number of times without ever returning to the identity element.

What is a cyclic group?

A cyclic group is a group in which every element can be generated by a single element. In other words, every element in the group belongs to a cyclic subgroup.

How can cyclic subgroups be used in mathematical proofs?

Cyclic subgroups can be useful in mathematical proofs because they often have well-defined properties that can be used to simplify calculations or prove certain statements. For example, if a group is known to be cyclic, then its order (number of elements) can be easily determined.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
693
  • Math POTW for University Students
Replies
1
Views
482
  • General Math
Replies
1
Views
700
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
719
  • Linear and Abstract Algebra
Replies
1
Views
616
Replies
2
Views
818
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
  • General Math
Replies
1
Views
1K
Back
Top