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In group theory, the single most important piece of information about a group is its cardinality, which is of course either finite, countably infinite, or uncountably infinite. Usually, however, uncountably infinite simply means a cardinality of $\aleph_{1}$, the same as $\mathbb{R}$. My question is: is there anywhere that groups with cardinality strictly greater than $\aleph_{1}$ arise naturally? Of course, it is easy enough to construct groups with arbitrarily large cardinality, but I cannot recall ever seeing them used.

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    $\begingroup$ The cardinality of $\mathbb{R}$ is not necessarily $\aleph_1$. $\endgroup$ Aug 13, 2010 at 0:45
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    $\begingroup$ What gave you the impression that the single most important piece of information about a group is its cardinality? It misses nearly all of the richness of group theory. For instance, many interesting classes of finite simple groups can be studied using a lot of the same tools as are used in the study of simple complex Lie algebras and semisimple linear algebraic groups. $\endgroup$
    – BCnrd
    Aug 13, 2010 at 0:54
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    $\begingroup$ Turning this remark around, can one give a "natural" example of a group which has cardinality $\aleph_1$, independent of CH? $\endgroup$ Aug 13, 2010 at 0:55
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    $\begingroup$ Pete, very good question! Here is one answer: it is consistent with ZFC that there are no Borel sets, and even no analytic sets, of size $\aleph_1$. In such a model of set theory, the only groups (built out of real numbers) of size $\aleph_1$ would have high descriptive set-theoretic complexity. This could be taken as a negative answer to your question. But a positive answer could still arise by builiding a group directly out of the countable ordinals, rather than by using reals. $\endgroup$ Aug 13, 2010 at 1:05
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    $\begingroup$ You can push it up to $\Sigma^1_2$ using the Mansfield Solovay theorem, and if you assume PD, then there are no projective sets of size $\aleph_1$. In those models, there are arguably no natural examples of sets of reals of size exactly $\aleph_1$. $\endgroup$ Aug 13, 2010 at 3:48

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In line with Joel's answer, my favorite "outrageously large group" is the group $G = \operatorname{Aut}(\mathbb{C})$ of field automorphisms of the complex numbers. It has cardinality $2^{2^{\aleph_0}}$, which is pretty scary. But that's just the beginning of how large it is. For instance, from the study of real-closed fields, one can deduce that the number of conjugacy classes of order $2$ elements of $G$ is also $2^{2^{\aleph_0}}$. It is also an extension of the absolute Galois group of $\mathbb{Q}$ (a profinite group which is conjectured to have among its quotients every finite group, up to isomorphism) by the huge simple group $\operatorname{Aut}(\mathbb{C}/\overline{\mathbb{Q}})$.

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    $\begingroup$ A nice reference for some of the basic facts on Aut(C) is the 1957 Lester R. Ford prize-winning article: Paul B. Yale, Automorphisms of the complex numbers, Math. Mag. 39 (1966), 135-141. jstor.org/stable/2689301 $\endgroup$
    – Skip
    Aug 13, 2010 at 0:59
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    $\begingroup$ Why is $\operatorname{Aut}(\mathbb C/\overline{\mathbb Q})$ simple? $\endgroup$ Mar 1, 2011 at 17:26
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    $\begingroup$ @Emil: That is a special case of a theorem of Daniel Lascar: see the end of math.uga.edu/~pete/galois.pdf for a precise reference. $\endgroup$ Mar 1, 2011 at 17:30
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I would expect that automorphism groups of natural structures would count as natural groups in your sense. But automorphism groups of uncountable structures often have size larger than the continuum. In general, the size of the automorphism group of a structure of size $\kappa$ is bounded above by $2^\kappa$, which is strictly larger than $\kappa$, and this upper bound is often reached, when the structure is insufficient to restrict the general nature of automorphisms. For example, the number of bijections of an infinite set of size $\kappa$ with itself is $2^\kappa$.

I am sure that you will be able to construct many other natural structures of uncountable size $\kappa$, whose automorphism groups have size $2^\kappa$, and these would seem to the sort of examples you seek.


P.S. Let me also note that your remark that the reals have size $\aleph_1$ is only correct when the Continuum Hypothesis holds. In general, the size of the reals, also known as the continuum, is $2^{\aleph_0}$, which is also denoted $\beth_1$, whereas $\aleph_1$ is simply the first uncountable cardinal.

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Does a group showing up in a College Algebra (pre-calculus) course count as arising naturally? I'm pretty sure we teach students to add two functions (from the reals to the reals) pointwise to get a new function, even there. Of course, on the one hand really we only ask them to deal with the countable subset of functions with a finite description, and on the other hand Abelian groups are not as interesting, but technically that defines a group with cardinality greater than the continuum. (We also define inverse functions and composition, but at first glance it seems that strictly monotone functions must have only continuum cardinality.)

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  • $\begingroup$ But if you are truly in pre-calculus, you likely only find piece-wise continuous functions, and even if you allow countably many pieces, you still have only continuum many such functions, the same size as $\mathbb{R}$, since on each piece they are determined by their values on the rationals. $\endgroup$ Aug 13, 2010 at 1:38
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    $\begingroup$ @Joel: Hence my caveat about only asking them to grok the countable subgroup of finitely described functions. $\endgroup$
    – Tracy Hall
    Aug 13, 2010 at 1:53
  • $\begingroup$ Yes, that is a nice way to say it. (+1). $\endgroup$ Aug 13, 2010 at 1:55
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    $\begingroup$ Every Banach space is an Abelian group, and many of the usual Banach spaces we study have larger cardinality than the reals. $\endgroup$ Aug 13, 2010 at 12:55
  • $\begingroup$ @CM: Really? (Meaning: "I know very little about Banach spaces", not "I doubt that very much.") What is such an example? $\endgroup$ Aug 13, 2010 at 23:24
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Any product group like $\{0, 1\}^I$ for index sets $I$, using mod 2 addition coordinatewise. It's just (isomorphic to) the power set of $I$ using symmetric difference as the addition. It's of course also a ring (pointwise multiplication / intersection ). These Boolean groups often come up in general topology.

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