[math-fun] more groups than rings, surprise
The Dec 2020 issue of the Amer Math Monthly has an article Desmond MacHale, Are there more finite rings than finite groups?, Amer. Math. Monthly, 127:10 (2020), 936-938. showing that there are infinitely many n such that there are more groups with n elements than rings with n elements. He gives one example, n = 36355, and asks if this is the smallest such n. This is a sequence not yet in the OEIS! Anyone care to investigate? These are the n such that A000001(n) > A027623(n).
Surely we could at least calculate A027623(32). I mean, the stupidest imaginable enumerator would take only a few minutes to run ... wouldn't it? While we are here -- I am as big a fan as anybody of not excluding the zero case whenever possible. But this case just seems weird to me. We have two sequences, A027623 and A037234, which differ only in the zero entry, which is 1 for the former and 0 for the latter. Clearly there must be angels-dancing-on-pins arguments lurking here, but I can't think of any. There aren't any rings with no elements, are there? On Tue, Jan 5, 2021 at 10:45 AM Neil Sloane <njasloane@gmail.com> wrote:
The Dec 2020 issue of the Amer Math Monthly has an article
Desmond MacHale, Are there more finite rings than finite groups?, Amer. Math. Monthly, 127:10 (2020), 936-938.
showing that there are infinitely many n such that there are more groups with n elements than rings with n elements. He gives one example, n = 36355, and asks if this is the smallest such n. This is a sequence not yet in the OEIS! Anyone care to investigate?
These are the n such that A000001(n) > A027623(n). _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Le mar. 5 janv. 2021 à 20:30, Allan Wechsler <acwacw@gmail.com> a écrit :
While we are here -- I am as big a fan as anybody of not excluding the zero case whenever possible. But this case just seems weird to me. We have two sequences, A027623 and A037234, which differ only in the zero entry, which is 1 for the former and 0 for the latter. Clearly there must be angels-dancing-on-pins arguments lurking here, but I can't think of any. There aren't any rings with no elements, are there?
I totally agree, AFAIK (and as the description of that sequence explicitly states) a ring is an abelian group (R,+) with a second law * etc., and a group must have a neutral element (which would then be the 0 of the ring). (See also A000001(0) = 0 : there are 0 groups with 0 elements.) I have no idea why one would let A027623(0) = 1. There are clearly 0 rings with 0 elements, since a ring has at least one element, the zero. If there was a ring with 0 elements, then forgetting about the multiplication we would have an abelian group with 0 elements, and thus A000001(0) >= 1. Anyway, the sequence's definition treats a(0)=1 as a special case: It does not claim that there is a ring with 0 elements. ("Number of rings..." is the definition of the terms a(n) with n >= 1, only.) The only suspicion of an explanation I can imagine is that this strange a(0)=1 was (erroneously) introduced because of the phrase "does not need to have a 1" (like, "we don't need a 1, so we can have a ring with only 0" - but of course, not 0 elements!). As a side note, f(0) = 1 is also in contradiction with the "mult" keyword saying that this is a multiplicative function, i.e., f(x y) = f(x) f(y) when gcd(x,y)=1. Considering y=0, this implies f(x)=1 for all x, if f(0)=1. (Usually multiplicative functions aren't even defined for index/argument 0, but if they are, either the case xy=0 must be excluded in the above equation, or f(0) must be equal to 0.) - Maximilian
On Tue, Jan 5, 2021 at 5:30 PM Allan Wechsler <acwacw@gmail.com> wrote:
There aren't any rings with no elements, are there?
Not as traditionally defined, but we can use a definition derived from the idea of a group with no elements to define a notion of a ring with no elements. https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html If we say a ring is a set R equipped with an associative commutative binary operation +: R x R -> R, a binary operation -: R x R -> R such that ∀g, h. g + (h - g) = h ∀g, h. (g + h) - g = h and a binary operation *: R x R -> R such that ∀g, h, j. g * (h + j) = g * h + g * j and ∀g, h, j. (h + j) * g = h * g + j * g, then when R is nonempty, it must have an additive unit given by 0 = g - g for any g. But the definition works just fine for the empty R as well. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
I would find the definition more compelling if it didn't have the condition "when R is nonempty, then". (E.g., the definition of a power set holds for all sets, including all the ones without elements.) —Dan
On Tuesday/5January/2021, at 7:34 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Jan 5, 2021 at 5:30 PM Allan Wechsler <acwacw@gmail.com <mailto:acwacw@gmail.com>> wrote:
There aren't any rings with no elements, are there?
Not as traditionally defined, but we can use a definition derived from the idea of a group with no elements to define a notion of a ring with no elements. https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html <https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html>
If we say a ring is a set R equipped with an associative commutative binary operation +: R x R -> R, a binary operation -: R x R -> R such that ∀g, h. g + (h - g) = h ∀g, h. (g + h) - g = h and a binary operation *: R x R -> R such that ∀g, h, j. g * (h + j) = g * h + g * j and ∀g, h, j. (h + j) * g = h * g + j * g,
then when R is nonempty, it must have an additive unit given by 0 = g - g for any g. But the definition works just fine for the empty R as well.
On Tue, Jan 5, 2021 at 10:56 PM Dan Asimov <asimov@msri.org> wrote:
I would find the definition more compelling if it didn't have the condition "when R is nonempty, then".
I think you misunderstand; the definition doesn't contain the phrase you mention above. The definition is just:
a ring is a set R equipped with an associative commutative binary operation +: R x R -> R, a binary operation -: R x R -> R such that ∀g, h. g + (h - g) = h ∀g, h. (g + h) - g = h and a binary operation *: R x R -> R such that ∀g, h, j. g * (h + j) = g * h + g * j and ∀g, h, j. (h + j) * g = h * g + j * g,
It is then a theorem that:
when R is nonempty, it has an additive unit given by 0 = g.
This definition is almost equivalent to the standard definition, the only difference being that this definition allows the empty set (with the obvious definitions of addition and multiplication) to be considered a ring.. Andy
(E.g., the definition of a power set holds for all sets, including all the ones without elements.)
—Dan
On Tuesday/5January/2021, at 7:34 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Jan 5, 2021 at 5:30 PM Allan Wechsler <acwacw@gmail.com <mailto: acwacw@gmail.com>> wrote:
There aren't any rings with no elements, are there?
Not as traditionally defined, but we can use a definition derived from the idea of a group with no elements to define a notion of a ring with no elements.
https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html < https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html
If we say a ring is a set R equipped with an associative commutative binary operation +: R x R -> R, a binary operation -: R x R -> R such that ∀g, h. g + (h - g) = h ∀g, h. (g + h) - g = h and a binary operation *: R x R -> R such that ∀g, h, j. g * (h + j) = g * h + g * j and ∀g, h, j. (h + j) * g = h * g + j * g,
then> when R is nonempty, it must have an additive unit given by 0 = g - g for any g. But the definition works just fine for the empty R as well.
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-- Andy.Latto@pobox.com
there are infinitely many n such that there are more groups with n elements than rings with n elements
Hm, I'm woefully uneducated in this stuff, but for this to even be an interesting question I think this says there must be groups that can be extended to rings in more than one way (since every ring an additive group augmented with a multiplicative operator). That suggests a table for the OEIS: T[n,k] = number of groups of order n that can be extended to (at most) k different rings. T[n,0] would be the number of groups of order n that cannot be augmented to be rings. Row sum over k of T[n,k] = number of groups of order n. Row sum over k of k * T[n,k] = number of rings of order n I'm curious: what is the smallest group G that supports more than one ring? (ie a G with min n such that T[n, some k > 1] is > 0)
BTW, he isn't assuming the ring has a unit, and IIRC he says that much less is known about that case. In case someone is going to program this. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Jan 5, 2021 at 8:49 PM Marc LeBrun <mlb@well.com> wrote:
there are infinitely many n such that there are more groups with n elements than rings with n elements
Hm, I'm woefully uneducated in this stuff, but for this to even be an interesting question I think this says there must be groups that can be extended to rings in more than one way (since every ring an additive group augmented with a multiplicative operator).
That suggests a table for the OEIS: T[n,k] = number of groups of order n that can be extended to (at most) k different rings.
T[n,0] would be the number of groups of order n that cannot be augmented to be rings.
Row sum over k of T[n,k] = number of groups of order n.
Row sum over k of k * T[n,k] = number of rings of order n
I'm curious: what is the smallest group G that supports more than one ring? (ie a G with min n such that T[n, some k > 1] is > 0)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It's easy to get tripped up by the terms of art. Apparently there is a word for a structure that is like a ring but is not required to have a multiplicative identity: a rng (pronounced "rung"). But that hasn't stopped some mathematicians from using "ring" for this concept, saying "ring with unit" when they mean "ring". Alas, this means that *every* time OEIS talks about rings, we have to clarify whether we mean to include rngs. A027623 is quite explicit and exemplary. On Tue, Jan 5, 2021 at 9:06 PM Neil Sloane <njasloane@gmail.com> wrote:
BTW, he isn't assuming the ring has a unit, and IIRC he says that much less is known about that case.
In case someone is going to program this.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Jan 5, 2021 at 8:49 PM Marc LeBrun <mlb@well.com> wrote:
there are infinitely many n such that there are more groups with n elements than rings with n elements
Hm, I'm woefully uneducated in this stuff, but for this to even be an interesting question I think this says there must be groups that can be extended to rings in more than one way (since every ring an additive group augmented with a multiplicative operator).
That suggests a table for the OEIS: T[n,k] = number of groups of order n that can be extended to (at most) k different rings.
T[n,0] would be the number of groups of order n that cannot be augmented to be rings.
Row sum over k of T[n,k] = number of groups of order n.
Row sum over k of k * T[n,k] = number of rings of order n
I'm curious: what is the smallest group G that supports more than one ring? (ie a G with min n such that T[n, some k > 1] is > 0)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Oh, regarding Marc LeBrun's interesting speculations on the number of rings (or rngs) that can be built with a given additive structure: remember that addition is required to be commutative in rngs. Are there any abelian groups that support no rngs? On Tue, Jan 5, 2021 at 9:18 PM Allan Wechsler <acwacw@gmail.com> wrote:
It's easy to get tripped up by the terms of art. Apparently there is a word for a structure that is like a ring but is not required to have a multiplicative identity: a rng (pronounced "rung"). But that hasn't stopped some mathematicians from using "ring" for this concept, saying "ring with unit" when they mean "ring".
Alas, this means that *every* time OEIS talks about rings, we have to clarify whether we mean to include rngs. A027623 is quite explicit and exemplary.
On Tue, Jan 5, 2021 at 9:06 PM Neil Sloane <njasloane@gmail.com> wrote:
BTW, he isn't assuming the ring has a unit, and IIRC he says that much less is known about that case.
In case someone is going to program this.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Jan 5, 2021 at 8:49 PM Marc LeBrun <mlb@well.com> wrote:
there are infinitely many n such that there are more groups with n elements than rings with n elements
Hm, I'm woefully uneducated in this stuff, but for this to even be an interesting question I think this says there must be groups that can be extended to rings in more than one way (since every ring an additive group augmented with a multiplicative operator).
That suggests a table for the OEIS: T[n,k] = number of groups of order n that can be extended to (at most) k different rings.
T[n,0] would be the number of groups of order n that cannot be augmented to be rings.
Row sum over k of T[n,k] = number of groups of order n.
Row sum over k of k * T[n,k] = number of rings of order n
I'm curious: what is the smallest group G that supports more than one ring? (ie a G with min n such that T[n, some k > 1] is > 0)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Le mar. 5 janv. 2021 à 22:23, Allan Wechsler <acwacw@gmail.com> a écrit :
Oh, regarding Marc LeBrun's interesting speculations on the number of rings (or rngs) that can be built with a given additive structure:
remember that addition is required to be commutative in rngs.
Are there any abelian groups that support no rngs?
No: you can always define x*y = 0 for all x,y, and get a ring (without unit, what some call a rng... ;-)) - Maximilian
I have a fair number of books on ring theory, and the term "ring" universally means a ring that may or may not have a 1. (That is also the convention in the OEIS.) No need to invent a Hebrew-style vowel-less name! The books by T Y Lam are the classics. There was a time when codes over rings (rather than fields) were all the rage. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Jan 5, 2021 at 9:23 PM Allan Wechsler <acwacw@gmail.com> wrote:
Oh, regarding Marc LeBrun's interesting speculations on the number of rings (or rngs) that can be built with a given additive structure: remember that addition is required to be commutative in rngs.
Are there any abelian groups that support no rngs?
On Tue, Jan 5, 2021 at 9:18 PM Allan Wechsler <acwacw@gmail.com> wrote:
It's easy to get tripped up by the terms of art. Apparently there is a word for a structure that is like a ring but is not required to have a multiplicative identity: a rng (pronounced "rung"). But that hasn't stopped some mathematicians from using "ring" for this concept, saying "ring with unit" when they mean "ring".
Alas, this means that *every* time OEIS talks about rings, we have to clarify whether we mean to include rngs. A027623 is quite explicit and exemplary.
On Tue, Jan 5, 2021 at 9:06 PM Neil Sloane <njasloane@gmail.com> wrote:
BTW, he isn't assuming the ring has a unit, and IIRC he says that much less is known about that case.
In case someone is going to program this.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Tue, Jan 5, 2021 at 8:49 PM Marc LeBrun <mlb@well.com> wrote:
there are infinitely many n such that there are more groups with n elements than rings with n elements
Hm, I'm woefully uneducated in this stuff, but for this to even be an interesting question I think this says there must be groups that can be extended to rings in more than one way (since every ring an additive group augmented with a multiplicative operator).
That suggests a table for the OEIS: T[n,k] = number of groups of order n that can be extended to (at most) k different rings.
T[n,0] would be the number of groups of order n that cannot be augmented to be rings.
Row sum over k of T[n,k] = number of groups of order n.
Row sum over k of k * T[n,k] = number of rings of order n
I'm curious: what is the smallest group G that supports more than one ring? (ie a G with min n such that T[n, some k > 1] is > 0)
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participants (7)
-
Allan Wechsler -
Andy Latto -
Dan Asimov -
M F Hasler -
Marc LeBrun -
Mike Stay -
Neil Sloane