Does Order Matter in Cyclic Subgroups/Groups?

[Gear300]

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>)?

 

[CompuChip]

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}>

 

[nirax]

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.

 

[Gear300]

I see...thanks for the replies.

 

[CompuChip]

Whoops, thanks nirax. Was already wondering why that sounded so odd ;)