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supernatural number
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(Definition)
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A supernatural number $\omega$ is a formal product
where $p$ runs over all (rational) prime numbers, and the values $n_p$ are each either natural numbers or the symbol $\infty$ .
We note first that by the fundamental theorem of arithmetic, we can view any natural number as a supernatural number. Supernatural numbers form a generalization of natural numbers in two ways: First, by allowing the possibility of infinitely many prime factors, and second, by allowing any given prime to divide $\omega$ ``infinitely often,'' by taking that prime's corresponding exponent to be the symbol $\infty$ .
We can extend the usual $p$ -adic order functions to these supernatural numbers by defining, for $\omega$ as above, $v_p(\omega)=n_p$ for each $p$ . We can then extend the notion of divisibility to supernatural numbers by declaring $\omega_1\mid\omega_2$ if $v_p(\omega_1)\leq v_p(\omega_2)$ for all $p$ (where, by definition, the symbol $\infty$ is considered greater than any natural number). Finally, we can also generalize the notion of the least common multiple (lcm) and greatest
common divisor (gcd) for (arbitrarily many) supernatural numbers, by defining
Note that the supernatural version of the definitions of divisibility, $\operatorname{lcm}$ , and $\operatorname{gcd}$ carry over exactly from their corresponding notions for natural numbers, though we can now take the gcd or lcm of infinitely many natural numbers (to get a supernatural number).
Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over verbatim.
- 1
- [Ram] Ramakrishnan, Dinikara and Valenza, Robert. Fourier Analysis on Number Fields. Graduate Texts in Mathematics, volume 186. Springer-Verlag, New York, NY. 1989.
- 2
- [Ser] Serre, J.-P. (Ion, P., translator) Galois Cohomology. Springer, New York, NY. 1997
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"supernatural number" is owned by mathcam.
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| Also defines: |
gcd of supernatural numbers, greatest commond divisor of supernatural numbers, lcm of supernatural numbers, least common multiple of supernatural numbers |
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Cross-references: theory, finite group, theorems, subgroups, profinite groups, indices, orders, definitions, greatest common divisor, least common multiple, divisibility, exponent, divide, prime factors, fundamental theorem of arithmetic, natural numbers, prime numbers, rational, product
There is 1 reference to this entry.
This is version 4 of supernatural number, born on 2005-07-15, modified 2006-08-10.
Object id is 7227, canonical name is SupernaturalNumber.
Accessed 7947 times total.
Classification:
| AMS MSC: | 20E18 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Limits, profinite groups) |
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Pending Errata and Addenda
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