Cyclic Subgroups/Groups

  • Thread starter 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>)?
 

CompuChip

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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}>
 
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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.
 
1,166
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I see...thanks for the replies.
 

CompuChip

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

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