Wikipedia:Reference desk/Archives/Mathematics/2015 February 11

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February 11

Name of a group

Is there a specific name for the group formed as (Q mod 1, +) = (Q/Z, +)? I was thinking of two ways of getting an infinite group as a limit from the sequence of finite cyclic groups Z_n = (Z/nZ, +). The first is simply to increase n uniformly, and to keep 1 as the generator, and the group becomes an open group, (Z, +). The second is to regard the same sequence of groups as the multiplicative group of the nth complex roots of unity C_n as n goes to infinity (but taking the limit in a special way, so that the resulting limit contains every finite cyclic group as a subgroup); we seem to end up with (exp(2πiQ), ×), but this is not cyclic in the normal sense of having one generator. —Quondum 00:27, 11 February 2015 (UTC)

I would call it the rational circle group. If you take a the limit of the cyclic groups with respect to the profinite topology, you get a different compact group which is the profinite completion of the integers. This is important in number theory because it is the Galois group of an algebraic closure of a finite field, as well as appearing in many other situations there, e.g., because of Artin reciprocity, in the ring of adeles, etc. A lot of modern number theory seems to concern in what sense this, and related objects, are "like" the usual circle that is associated with classical Fourier analysis that one gets by taking "the usual" completion of the rational points on the circle (to get the standard thing we would call "circle"). Sławomir Biały (talk) 01:03, 11 February 2015 (UTC)

Thanks. I see that the name pops up in a Google search (not often, mind you). It seems common to refer to it as the (additive) quotient group Q/Z. The other related groups concepts you linked to have me drowning in terminology new to me. I stumbled across the group of rational points on the unit circle, another group with similar properties: also a subgroup of the circle group and with the same cardinality, but with a very different structure; (C₄ and its subgroups are possibly the only shared subgroups. Something for me to think about a bit. —Quondum 05:27, 11 February 2015 (UTC)

Operators in Finite Groups

Is each finite group defined only for one specific operator? For example, the cyclic group with 7 elements has the operator addition modulo 7. Does this mean that any other 7 element group will have an operator that is isomorphic to addition modulo 7? Or is there a 7 element group with a different operator? OldTimeNESter (talk) 13:19, 11 February 2015 (UTC)

Any two groups of order 7 are isomorphic, because 7 is prime. The same is not true for any composite integer. For example, there are two groups of order 6: one is a cyclic group (rotations of a hexagon) and the other is a dihedral group (rotations and reflections of a triangle). If you restrict attention just to abelian groups, then any abelian group whose order is square-free is cyclic, but otherwise (if a square of a prime divides the order) there is the cyclic group in that order, and products of cyclic groups that will be non-isomorphic. Sławomir Biały (talk) 13:31, 11 February 2015 (UTC)

The article List of small groups can be instructive in this regard. —Quondum 14:36, 11 February 2015 (UTC)

The group whose elements are 7th roots of unity in the complex plane is a standard example of a cyclic group of order 7. Here the group operation is multiplication of complex numbers of modulus 1. Naturally, this works for any finite order. YohanN7 (talk) 15:48, 11 February 2015 (UTC)