Timeline for Countable group with uncountable number of subgroups $< 2^{\aleph_0}$

5 events

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Jun 18, 2015 at 12:50	comment	added	Emil Jeřábek		@Goldstern: Indeed. Though I gather that in case of a simple question like here, the only nonobvious part is to realize it is connected to topology in the first place; once you know that, you already have the answer.
Jun 18, 2015 at 11:54	comment	added	Goldstern		I think that questions of the form "Does $\|X\|\ge 2^{\aleph_0}$, i.e., is there a 1-1 map from the Cantor space into $X$?" should always be expanded to "Does $X$ contain a perfect set in some natural T2 topology", or equivalently "Is there a CONTINUOUS 1-1 map from the Cantor space into $X$?". It turns out that a ZFC-answer of "yes" to the first question almost always (in particular: here) is proved by showing that even the second question has a positive answer. The perfect set answer is more interesting because it shows a structural result about $X$, more than a mere cardinality estimate.
Jun 18, 2015 at 6:38	vote	accept	Dominic van der Zypen
Jun 18, 2015 at 1:39	comment	added	The Masked Avenger		There is an exercise attributed to Burris and Kwatinetz at the end of Chapter 1 in the book "Algebras, Lattices, Varieties Volume I" which mentions (for countable algebras of countable type) similar results on the size of the automorphism group and endomorphism monoid and congruence lattice as well.
Jun 17, 2015 at 15:17	history	answered	Emil Jeřábek	CC BY-SA 3.0