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In [[group theory]], a '''locally cyclic group''' is a group (''G'', *) in which every [[generating set of a group|finitely generated subgroup]] is [[cyclic group|cyclic]].
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==Some facts==
*Every cyclic group is locally cyclic, and every locally cyclic group is [[abelian group|abelian]].
*Every finitely-generated locally cyclic group is cyclic.
*Every [[subgroup]] and [[quotient group]] of a locally cyclic group is locally cyclic.
*Every [[homomorphism|Homomorphic]] image of a locally cyclic group is locally cyclic.
*A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
*A group is locally cyclic if and only if its [[lattice of subgroups]] is [[distributive lattice|distributive]] {{harv|Ore|1938}}.
*The [[torsion-free rank]] of a locally cyclic group is 0 or 1.
 
==Examples of locally cyclic groups that are not cyclic==
*The additive group of [[rational number]]s ('''Q''', +) is locally cyclic – any pair of rational numbers ''a''/''b'' and ''c''/''d'' is contained in the cyclic subgroup generated by 1/''bd''.
*The additive group of the [[dyadic rational number]]s, the rational numbers of the form ''a''/2<sup>''b''</sup>, is also locally cyclic – any pair of dyadic rational numbers ''a''/2<sup>''b''</sup> and ''c''/2<sup>''d''</sup> is contained in the cyclic subgroup generated by 1/2<sup>max(''b'',''d'')</sup>.
*Let ''p'' be any prime, and let μ<sub>''p''<sup>∞</sup></sub> denote the set of all ''p''th-power [[root of unity|roots of unity]] in '''C''', i.e.
 
:<math>\mu_{p^{\infty}} = \left\{ \exp\left(\frac{2\pi im}{p^{k}}\right) : m,k\in\mathbb{Z}\right\}</math>
 
:Then &mu;<sub>''p''<sup>&infin;</sup></sub> is locally cyclic but not cyclic. This is the [[Prüfer group|Prüfer ''p''-group]]. The Prüfer 2-group is closely related to the dyadic rationals (it can be viewed as the dyadic rationals modulo 1).
 
==Examples of abelian groups that are not locally cyclic==
*The additive group of [[real number]]s ('''R''', +) is not locally cyclic—the subgroup generated by 1 and π consists of all numbers of the form ''a'' + ''b''π. This group is [[group isomorphism|isomorphic]] to the [[direct sum of groups|direct sum]] '''Z''' + '''Z''', and this group is not cyclic.
 
==References==
*{{citation
| last = Hall | first = Marshall, Jr. | author-link = Marshall Hall (mathematician)
| contribution = 19.2 Locally Cyclic Groups and Distributive Lattices
| isbn = 978-0-8218-1967-8
| pages = 340–341
| publisher = American Mathematical Society
| title = Theory of Groups
| year = 1999}}.
 
*{{citation
| last = Ore | first = Øystein | author-link = Øystein Ore
| doi = 10.1215/S0012-7094-38-00419-3
| mr = 1546048
| issue = 2
| journal = Duke Mathematical Journal
| pages = 247–269
| title = Structures and group theory. II
| volume = 4
| year = 1938}}.
 
[[Category:Abelian group theory]]

Revision as of 06:21, 23 February 2014

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To start with, you must know whether or not the UGG boots mix sheepskin and fur alongside one another, combining the sheepskin and fur jointly indicates that the sheepskin with fur is stripped from the sheep, after the creation, one particular aspect of the sheepskin is the inside of of boots, the other aspect of the sheepskin is the upper of the boots, you can differentiate this from the crevice. You can also drag the sheepskin, the sheepskin will not be pulled down. The fur r is on the leather, if you glimpse thoroughly, you will locate that the fur grows from the pores, instead than caught to the pore, so the fur will not be pull down. In addition, the dermal layer itself is divided into the sub-epidermis and pores and skin corium, you can see the boundary and a minor dot of hair follicles. If the boots do not mix the sheepskin and fur collectively, then the fur is woven on a layer of mesh, you can definitely see the guidelines of grid from the base. Regardless of whether the UGG 5815 or the UGG 5825 chestnut leopard, you can check out them in this way.
Next, you can check the wool thickness of UGG snow boots, the good quality of Australian sheep pores and skin is much greater than the domestic ones, there is no question, and the wool within the authentic UGG boots is at the very least one cm thick, and the wool is extremely neat. Eventually, you can seem at the bottom of the boots, the genuine UGG snow boots use rubber foam, it appears to be like like the slipper foam sole, but you can come across that it is the rubber sole at a nearer glimpse. One more sole is identical to the outdoor sole of the climbing footwear, it is a lot a lot more highly-priced than the rubber foam sole, and they surely have the UGG emblem.
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