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Is there a standard notation for the multiplicative group generated by the primes $pin P$?
Clash Royale CLAN TAG#URR8PPP
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Is there a standard notation for the multiplicative group generated by the primes $pin P$?
Let $P$ be some set of primes e.g. $P=2,3$
Then $G_P$ is the multiplicative group generated by these primes so e.g. $G_2,3$ is the 3-smooth numbers and their inverses, with multiplication.
Is there a standard notation or way of expressing this group and similar?
group-theory notation
asked Jul 27 at 17:42
Robert Frost
3,884936
add a comment |Â
up vote
1
down vote
favorite
Is there a standard notation for the multiplicative group generated by the primes $pin P$?
Let $P$ be some set of primes e.g. $P=2,3$
Then $G_P$ is the multiplicative group generated by these primes so e.g. $G_2,3$ is the 3-smooth numbers and their inverses, with multiplication.
Is there a standard notation or way of expressing this group and similar?
group-theory notation
asked Jul 27 at 17:42
Robert Frost
3,884936
2
Up to isomorphism this group is just a product of $|P|$ copies of $(BbbZ, 0, +)$, so one standard notation would be $BbbZ^P$..
– Rob Arthan
Jul 27 at 18:15
@RobArthan coolio, thanks. I was wrestling with $2,3^<omega$ and I knew it was wrong because in that, the orders of the sequences of primes would matter- but I couldn't get it straight but yours makes total sense.
– Robert Frost
Jul 27 at 18:16
@RobArthan yes, please do.
– Robert Frost
Jul 27 at 18:39
I've posted an answer with a bit of extra discussion.
– Rob Arthan
Jul 27 at 18:42
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Is there a standard notation for the multiplicative group generated by the primes $pin P$?
Let $P$ be some set of primes e.g. $P=2,3$
Then $G_P$ is the multiplicative group generated by these primes so e.g. $G_2,3$ is the 3-smooth numbers and their inverses, with multiplication.
Is there a standard notation or way of expressing this group and similar?
group-theory notation
asked Jul 27 at 17:42
Robert Frost
3,884936
Is there a standard notation for the multiplicative group generated by the primes $pin P$?
Let $P$ be some set of primes e.g. $P=2,3$
Then $G_P$ is the multiplicative group generated by these primes so e.g. $G_2,3$ is the 3-smooth numbers and their inverses, with multiplication.
Is there a standard notation or way of expressing this group and similar?
group-theory notation
asked Jul 27 at 17:42
Robert Frost
3,884936
asked Jul 27 at 17:42
Robert Frost
3,884936
asked Jul 27 at 17:42
Robert Frost
3,884936
asked Jul 27 at 17:42
Robert Frost
3,884936
3,884936
2
Up to isomorphism this group is just a product of $|P|$ copies of $(BbbZ, 0, +)$, so one standard notation would be $BbbZ^P$..
– Rob Arthan
Jul 27 at 18:15
@RobArthan coolio, thanks. I was wrestling with $2,3^<omega$ and I knew it was wrong because in that, the orders of the sequences of primes would matter- but I couldn't get it straight but yours makes total sense.
– Robert Frost
Jul 27 at 18:16
@RobArthan yes, please do.
– Robert Frost
Jul 27 at 18:39
I've posted an answer with a bit of extra discussion.
– Rob Arthan
Jul 27 at 18:42
add a comment |Â
2
Up to isomorphism this group is just a product of $|P|$ copies of $(BbbZ, 0, +)$, so one standard notation would be $BbbZ^P$..
– Rob Arthan
Jul 27 at 18:15
@RobArthan coolio, thanks. I was wrestling with $2,3^<omega$ and I knew it was wrong because in that, the orders of the sequences of primes would matter- but I couldn't get it straight but yours makes total sense.
– Robert Frost
Jul 27 at 18:16
@RobArthan yes, please do.
– Robert Frost
Jul 27 at 18:39
I've posted an answer with a bit of extra discussion.
– Rob Arthan
Jul 27 at 18:42
2
2
Up to isomorphism this group is just a product of $|P|$ copies of $(BbbZ, 0, +)$, so one standard notation would be $BbbZ^P$..
– Rob Arthan
Jul 27 at 18:15
Up to isomorphism this group is just a product of $|P|$ copies of $(BbbZ, 0, +)$, so one standard notation would be $BbbZ^P$..
– Rob Arthan
Jul 27 at 18:15
@RobArthan coolio, thanks. I was wrestling with $2,3^<omega$ and I knew it was wrong because in that, the orders of the sequences of primes would matter- but I couldn't get it straight but yours makes total sense.
– Robert Frost
Jul 27 at 18:16
@RobArthan coolio, thanks. I was wrestling with $2,3^<omega$ and I knew it was wrong because in that, the orders of the sequences of primes would matter- but I couldn't get it straight but yours makes total sense.
– Robert Frost
Jul 27 at 18:16
@RobArthan yes, please do.
– Robert Frost
Jul 27 at 18:39
@RobArthan yes, please do.
– Robert Frost
Jul 27 at 18:39
I've posted an answer with a bit of extra discussion.
– Rob Arthan
Jul 27 at 18:42
I've posted an answer with a bit of extra discussion.
– Rob Arthan
Jul 27 at 18:42
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
When $P = 2, 3$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j in BbbZ$ and this is easily checked to give an isomorphism between $G_P$ and the sum $BbbZ^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality $|P|$ of $P$, so one standard notation for $G_P$ is $Bbb~Z^P$.
This should be understood subject to the proviso that, if $P$ is infinite, $Bbb~Z^P$ is to be interpreted as the infinite sum and not the infinite product (i.e., it only includes sequences $(i_1, i_2, ldots)$ where all but finitely many of the $i_j$ are zero).
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
1
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
1
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
2
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
When $P = 2, 3$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j in BbbZ$ and this is easily checked to give an isomorphism between $G_P$ and the sum $BbbZ^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality $|P|$ of $P$, so one standard notation for $G_P$ is $Bbb~Z^P$.
This should be understood subject to the proviso that, if $P$ is infinite, $Bbb~Z^P$ is to be interpreted as the infinite sum and not the infinite product (i.e., it only includes sequences $(i_1, i_2, ldots)$ where all but finitely many of the $i_j$ are zero).
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
1
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
1
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
2
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
add a comment |Â
up vote
3
down vote
accepted
When $P = 2, 3$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j in BbbZ$ and this is easily checked to give an isomorphism between $G_P$ and the sum $BbbZ^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality $|P|$ of $P$, so one standard notation for $G_P$ is $Bbb~Z^P$.
This should be understood subject to the proviso that, if $P$ is infinite, $Bbb~Z^P$ is to be interpreted as the infinite sum and not the infinite product (i.e., it only includes sequences $(i_1, i_2, ldots)$ where all but finitely many of the $i_j$ are zero).
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
1
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
1
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
2
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
When $P = 2, 3$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j in BbbZ$ and this is easily checked to give an isomorphism between $G_P$ and the sum $BbbZ^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality $|P|$ of $P$, so one standard notation for $G_P$ is $Bbb~Z^P$.
This should be understood subject to the proviso that, if $P$ is infinite, $Bbb~Z^P$ is to be interpreted as the infinite sum and not the infinite product (i.e., it only includes sequences $(i_1, i_2, ldots)$ where all but finitely many of the $i_j$ are zero).
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
When $P = 2, 3$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j in BbbZ$ and this is easily checked to give an isomorphism between $G_P$ and the sum $BbbZ^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality $|P|$ of $P$, so one standard notation for $G_P$ is $Bbb~Z^P$.
This should be understood subject to the proviso that, if $P$ is infinite, $Bbb~Z^P$ is to be interpreted as the infinite sum and not the infinite product (i.e., it only includes sequences $(i_1, i_2, ldots)$ where all but finitely many of the $i_j$ are zero).
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
answered Jul 27 at 18:41
Rob Arthan
27.1k42863
27.1k42863
1
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
1
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
2
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
add a comment |Â
1
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
1
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
2
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
1
1
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
Per your last sentence, I have a slight preference for "$bigoplus_vert PvertmathbbZ$."
– Noah Schweber
Jul 27 at 19:00
1
1
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
@NoahSchweber: I'm inclined to agree with you. In an ideal world, perhaps we would use $BbbZ^kappa$ for the $kappa$-fold product and $kappa BbbZ$ for the $kappa$-fold sum. But that conflicts with the desire to write $2BbbZ$ for the set of even integers.
– Rob Arthan
Jul 27 at 20:31
2
2
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
This only names the abstract isomorphism class of $G_P$; you might also want to explicitly name $G_P$ as a subgroup of $mathbbQ$.
– Qiaochu Yuan
Jul 27 at 21:43
add a comment |Â
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2
Up to isomorphism this group is just a product of $|P|$ copies of $(BbbZ, 0, +)$, so one standard notation would be $BbbZ^P$..
– Rob Arthan
Jul 27 at 18:15
@RobArthan coolio, thanks. I was wrestling with $2,3^<omega$ and I knew it was wrong because in that, the orders of the sequences of primes would matter- but I couldn't get it straight but yours makes total sense.
– Robert Frost
Jul 27 at 18:16
@RobArthan yes, please do.
– Robert Frost
Jul 27 at 18:39
I've posted an answer with a bit of extra discussion.
– Rob Arthan
Jul 27 at 18:42