How do I multiply permutations? From left or from right?
$begingroup$
How do I multiply permutations? From left or from right? My understanding is that I go from right to left as the permutation closest to the element acts first:
- I assume it's $(12)(13)(14)=(1432)$
- Or is it really $(12)(13)(14)=(1234)$?
permutations convention
edited Dec 22 '18 at 12:42
Shaun
9,366113684
asked Nov 2 '17 at 17:07
BuhBuh
60827
$endgroup$
|
show 1 more comment
$begingroup$
How do I multiply permutations? From left or from right? My understanding is that I go from right to left as the permutation closest to the element acts first:
- I assume it's $(12)(13)(14)=(1432)$
- Or is it really $(12)(13)(14)=(1234)$?
permutations convention
edited Dec 22 '18 at 12:42
Shaun
9,366113684
asked Nov 2 '17 at 17:07
BuhBuh
60827
$endgroup$
3
$begingroup$
You can do it either way as long as you do it the same way all the time. Different books/classes make different decisions at the start of the exposition, then stick with it. I personally prefer (2).
$endgroup$
– Ethan Bolker
Nov 2 '17 at 17:10
$begingroup$
But doesn't that somehow conflict with $(fcirc g)(z)=f(g(z))$?
$endgroup$
– Buh
Nov 2 '17 at 17:17
$begingroup$
@Buh Some people also read function composition from left to right.
$endgroup$
– Qudit
Nov 2 '17 at 17:48
1
$begingroup$
@Buh Yes it conflicts with that interpretation, but when I'm working in the symmetric group I think of it more abstractly as products of cycles with left to right multiplication. If I need to interpret the permutations as functions then they are acting on the other side of their arguments. Herstein's algebra text writes function applications as $(x)f$ precisely so that composition reads left to right. I don't comfortably go that far.
$endgroup$
– Ethan Bolker
Nov 2 '17 at 18:30
$begingroup$
I see. Very interesting! Thank you for this answer.
$endgroup$
– Buh
Nov 2 '17 at 19:50
|
show 1 more comment
$begingroup$
How do I multiply permutations? From left or from right? My understanding is that I go from right to left as the permutation closest to the element acts first:
- I assume it's $(12)(13)(14)=(1432)$
- Or is it really $(12)(13)(14)=(1234)$?
permutations convention
edited Dec 22 '18 at 12:42
Shaun
9,366113684
asked Nov 2 '17 at 17:07
BuhBuh
60827
$endgroup$
How do I multiply permutations? From left or from right? My understanding is that I go from right to left as the permutation closest to the element acts first:
- I assume it's $(12)(13)(14)=(1432)$
- Or is it really $(12)(13)(14)=(1234)$?
permutations convention
permutations convention
edited Dec 22 '18 at 12:42
Shaun
9,366113684
asked Nov 2 '17 at 17:07
BuhBuh
60827
edited Dec 22 '18 at 12:42
Shaun
9,366113684
asked Nov 2 '17 at 17:07
BuhBuh
60827
edited Dec 22 '18 at 12:42
Shaun
9,366113684
edited Dec 22 '18 at 12:42
Shaun
9,366113684
edited Dec 22 '18 at 12:42
Shaun
9,366113684
9,366113684
asked Nov 2 '17 at 17:07
BuhBuh
60827
asked Nov 2 '17 at 17:07
BuhBuh
60827
asked Nov 2 '17 at 17:07
BuhBuh
60827
60827
3
$begingroup$
You can do it either way as long as you do it the same way all the time. Different books/classes make different decisions at the start of the exposition, then stick with it. I personally prefer (2).
$endgroup$
– Ethan Bolker
Nov 2 '17 at 17:10
$begingroup$
But doesn't that somehow conflict with $(fcirc g)(z)=f(g(z))$?
$endgroup$
– Buh
Nov 2 '17 at 17:17
$begingroup$
@Buh Some people also read function composition from left to right.
$endgroup$
– Qudit
Nov 2 '17 at 17:48
1
$begingroup$
@Buh Yes it conflicts with that interpretation, but when I'm working in the symmetric group I think of it more abstractly as products of cycles with left to right multiplication. If I need to interpret the permutations as functions then they are acting on the other side of their arguments. Herstein's algebra text writes function applications as $(x)f$ precisely so that composition reads left to right. I don't comfortably go that far.
$endgroup$
– Ethan Bolker
Nov 2 '17 at 18:30
$begingroup$
I see. Very interesting! Thank you for this answer.
$endgroup$
– Buh
Nov 2 '17 at 19:50
|
show 1 more comment
3
$begingroup$
You can do it either way as long as you do it the same way all the time. Different books/classes make different decisions at the start of the exposition, then stick with it. I personally prefer (2).
$endgroup$
– Ethan Bolker
Nov 2 '17 at 17:10
$begingroup$
But doesn't that somehow conflict with $(fcirc g)(z)=f(g(z))$?
$endgroup$
– Buh
Nov 2 '17 at 17:17
$begingroup$
@Buh Some people also read function composition from left to right.
$endgroup$
– Qudit
Nov 2 '17 at 17:48
1
$begingroup$
@Buh Yes it conflicts with that interpretation, but when I'm working in the symmetric group I think of it more abstractly as products of cycles with left to right multiplication. If I need to interpret the permutations as functions then they are acting on the other side of their arguments. Herstein's algebra text writes function applications as $(x)f$ precisely so that composition reads left to right. I don't comfortably go that far.
$endgroup$
– Ethan Bolker
Nov 2 '17 at 18:30
$begingroup$
I see. Very interesting! Thank you for this answer.
$endgroup$
– Buh
Nov 2 '17 at 19:50
3
3
$begingroup$
You can do it either way as long as you do it the same way all the time. Different books/classes make different decisions at the start of the exposition, then stick with it. I personally prefer (2).
$endgroup$
– Ethan Bolker
Nov 2 '17 at 17:10
$begingroup$
You can do it either way as long as you do it the same way all the time. Different books/classes make different decisions at the start of the exposition, then stick with it. I personally prefer (2).
$endgroup$
– Ethan Bolker
Nov 2 '17 at 17:10
$begingroup$
But doesn't that somehow conflict with $(fcirc g)(z)=f(g(z))$?
$endgroup$
– Buh
Nov 2 '17 at 17:17
$begingroup$
But doesn't that somehow conflict with $(fcirc g)(z)=f(g(z))$?
$endgroup$
– Buh
Nov 2 '17 at 17:17
$begingroup$
@Buh Some people also read function composition from left to right.
$endgroup$
– Qudit
Nov 2 '17 at 17:48
$begingroup$
@Buh Some people also read function composition from left to right.
$endgroup$
– Qudit
Nov 2 '17 at 17:48
1
1
$begingroup$
@Buh Yes it conflicts with that interpretation, but when I'm working in the symmetric group I think of it more abstractly as products of cycles with left to right multiplication. If I need to interpret the permutations as functions then they are acting on the other side of their arguments. Herstein's algebra text writes function applications as $(x)f$ precisely so that composition reads left to right. I don't comfortably go that far.
$endgroup$
– Ethan Bolker
Nov 2 '17 at 18:30
$begingroup$
@Buh Yes it conflicts with that interpretation, but when I'm working in the symmetric group I think of it more abstractly as products of cycles with left to right multiplication. If I need to interpret the permutations as functions then they are acting on the other side of their arguments. Herstein's algebra text writes function applications as $(x)f$ precisely so that composition reads left to right. I don't comfortably go that far.
$endgroup$
– Ethan Bolker
Nov 2 '17 at 18:30
$begingroup$
I see. Very interesting! Thank you for this answer.
$endgroup$
– Buh
Nov 2 '17 at 19:50
$begingroup$
I see. Very interesting! Thank you for this answer.
$endgroup$
– Buh
Nov 2 '17 at 19:50
|
show 1 more comment
1 Answer
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$begingroup$
The order of composition of permutations is a matter of convention. Both directions are in use.
However, since the argument of a function is traditionally on the right (and a permutation is a function on its defining set), it is more common to find the right-to-left convention.
Examples of when the left-to-right convention is in use are semigroup theory and formal languages.
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
$endgroup$
add a comment |
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$begingroup$
The order of composition of permutations is a matter of convention. Both directions are in use.
However, since the argument of a function is traditionally on the right (and a permutation is a function on its defining set), it is more common to find the right-to-left convention.
Examples of when the left-to-right convention is in use are semigroup theory and formal languages.
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
$endgroup$
add a comment |
$begingroup$
The order of composition of permutations is a matter of convention. Both directions are in use.
However, since the argument of a function is traditionally on the right (and a permutation is a function on its defining set), it is more common to find the right-to-left convention.
Examples of when the left-to-right convention is in use are semigroup theory and formal languages.
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
$endgroup$
add a comment |
$begingroup$
The order of composition of permutations is a matter of convention. Both directions are in use.
However, since the argument of a function is traditionally on the right (and a permutation is a function on its defining set), it is more common to find the right-to-left convention.
Examples of when the left-to-right convention is in use are semigroup theory and formal languages.
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
$endgroup$
The order of composition of permutations is a matter of convention. Both directions are in use.
However, since the argument of a function is traditionally on the right (and a permutation is a function on its defining set), it is more common to find the right-to-left convention.
Examples of when the left-to-right convention is in use are semigroup theory and formal languages.
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
answered Dec 22 '18 at 12:41
ShaunShaun
9,366113684
9,366113684
add a comment |
add a comment |
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3
$begingroup$
You can do it either way as long as you do it the same way all the time. Different books/classes make different decisions at the start of the exposition, then stick with it. I personally prefer (2).
$endgroup$
– Ethan Bolker
Nov 2 '17 at 17:10
$begingroup$
But doesn't that somehow conflict with $(fcirc g)(z)=f(g(z))$?
$endgroup$
– Buh
Nov 2 '17 at 17:17
$begingroup$
@Buh Some people also read function composition from left to right.
$endgroup$
– Qudit
Nov 2 '17 at 17:48
1
$begingroup$
@Buh Yes it conflicts with that interpretation, but when I'm working in the symmetric group I think of it more abstractly as products of cycles with left to right multiplication. If I need to interpret the permutations as functions then they are acting on the other side of their arguments. Herstein's algebra text writes function applications as $(x)f$ precisely so that composition reads left to right. I don't comfortably go that far.
$endgroup$
– Ethan Bolker
Nov 2 '17 at 18:30
$begingroup$
I see. Very interesting! Thank you for this answer.
$endgroup$
– Buh
Nov 2 '17 at 19:50