Primitive abundant number: Difference between revisions

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In Erdos's paper it is a small-o.
 
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In mathematics, specifically [[group theory]], '''isoclinism''' is an [[equivalence relation]] on [[group (mathematics)|groups]] which generalizes [[isomorphism]]. The concept of isoclinism was introduced by {{harvtxt|Hall|1940}} to help classify and understand [[p-group]]s, although applicable to all groups. Isoclinism remains an important part of the study of p-groups, and for instance §29 of {{harvtxt|Berkovich|2008}} and §21.2 of {{harvtxt|Blackburn|Neumann|Venkataraman|2007}} are devoted to it.  Isoclinism also has vital consequences<!--ATLAS p. xxiii, §6.7¶3--> for the [[Schur multiplier]] and the associated aspects of [[character theory]], as described in {{harvtxt|Suzuki|1982|p=256}} and {{harvtxt|Conway|Curtis|Norton|Parker|1985|p=xxiii|loc=Ch. 6.7}}.
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==Definition==
According to {{harvtxt|Struik|1960}}, two groups G<sub>1</sub> and G<sub>2</sub> are isoclinic if the following three conditions hold:
 
# <math>G_1/Z_1</math> is isomorphic to <math>G_2/Z_2</math>, where <math>Z_1</math> is the [[center (group theory)|center]] of <math>G_1</math> and <math>Z_2</math> is the center of <math>G_2</math>.
# The [[commutator subgroup]] of <math>G_1</math> is isomorphic to the commutator subgroup of <math>G_2</math>.
# "The isomorphisms of (1) and (2) can be selected in such a way that whenever <math>a_1Z_1</math> and <math>b_1Z_1</math> correspond respectively to <math>a_2Z_2</math> and <math>b_2Z_2</math> under (1), then <math>a_1b_1a_1^{-1}b_1^{-1}</math> corresponds to <math>a_2b_2a_2^{-1}b_2^{-1}</math> under (2)."
 
==Examples==
All [[Abelian group]]s are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and ''G'' is isoclinic with ''G''×''A'' if and only if ''A'' is abelian. The [[dihedral group|dihedral]], [[quasidihedral group|quasidihedral]], and [[quaternion group]]s of order 2<sup>''n''</sup> are isoclinic for ''n''≥3, {{harvtxt|Berkovich|2008|p=285}}.
 
Isoclinism divides ''p''-groups into families, and the smallest members of each family are called '''stem groups'''. A group is a stem group if and only if Z(''G'') ≤ [''G'',''G''], that is, if and only if every element of the center of the group is contained in the [[derived subgroup]] (also called the commutator subgroup), {{harvtxt|Berkovich|2008|p=287}}. Some enumeration results on isoclinism families are given in {{harvtxt|Blackburn|Neumann|Venkataraman|2007|p=226}}.
 
Another textbook treatment of isoclinism is given in {{harvtxt|Suzuki|1986|pp=92–95}}, which describes in more detail the isomorphisms induced by an isoclinism. Isoclinism is important in theory of [[projective representation]]s of [[finite group]]s, as all [[Schur cover]]ing groups of a group are isoclinic, a fact already hinted at by Hall according to {{harvtxt|Suzuki|1982|p=256}}. This is important in describing the character tables of the [[finite simple group]]s, and so is described in some detail in {{harvtxt|Conway|Curtis|Norton|Parker|1985|p=xxiii|loc=Ch. 6.7}}.
 
== References ==
* {{Citation | last1=Berkovich | first1=Yakov | title=Groups of prime power order. Vol. 1 | publisher=Walter de Gruyter GmbH & Co. KG, Berlin | series=de Gruyter Expositions in Mathematics | isbn=978-3-11-020418-6 | doi=10.1515/9783110208221.285 | id={{MathSciNet | id = 2464640}} | year=2008 | volume=46}}
*{{Citation | last1=Blackburn | first1=Simon R. | last2=Neumann | first2=Peter M. | author2-link=Peter M. Neumann | last3=Venkataraman | first3=Geetha | title=Enumeration of finite groups | publisher=[[Cambridge University Press]] | edition=1st | series=Cambridge Tracts in Mathematics no 173 | isbn=978-0-521-88217-0 | oclc=154682311 | year=2007}}
* {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Curtis | first2=R. T. | last3=Norton | first3=S. P. | last4=Parker | first4=R. A. | last5=Wilson | first5=R. A. | title=Atlas of finite groups | publisher=[[Oxford University Press]] | isbn=978-0-19-853199-9 | id={{MathSciNet | id = 827219}} | year=1985}}
*{{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=The classification of prime-power groups | doi=10.1515/crll.1940.182.130 | id={{MathSciNet | id = 0003389}} | year=1940 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=182 | pages=130–141 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00217491X }}
*{{Citation | last1=Struik | first1=Ruth Rebekka | title=A note on prime-power groups | url=http://math.ca/cmb/v3/p27 | id={{MathSciNet | id = 0148744}} | year=1960 | journal=[[Canadian Mathematical Bulletin]] | issn=0008-4395 | volume=3 | pages=27–30}}
*{{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=Group theory. I | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-10915-0 | id={{MathSciNet | id = 648772}} | year=1982 | volume=247}}
*{{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=Group theory. II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-0-387-10916-9 | id={{MathSciNet | id = 815926}} | year=1986 | volume=248}}
 
{{DEFAULTSORT:Isoclinism Of Groups}}
[[Category:Finite groups]]

Latest revision as of 14:04, 24 June 2014

The concept of the Christy knife is elegantly easy, and it ships with a key ring in a pleasant plastic reusable case. The blade slides in a channel contained in the handle and locks into one of four positions, including closed. In a time earlier than Spyderco had the idea to put a gap in a blade for thumb opening, this groundbreaking, single-handed working knife was easy to use by sliding the round spring loaded button down and forward or again to open or close the knife. After a sure point in a knife's life these do probably not sharpen so nicely and the knife blade must be sharpened.



The second can opener Kitchen Ect sells is the "Brushed Chrome Can Opener." This can opener can open any size can with its super sharp can opening blade with out slipping and sliding off the electric can opener. It additionally has somewhat lever that presses down while you wish to open a can and lifts up when you're performed opening the can, and is removable while you need to wash it up. It even can retailer the can opener's electrical twine at the back of it. The model quantity for this may opener is CCO-40BC and it value twenty dollars and has a three year warranty.

Different reasons I like these little knives are their mild weight, ease of use, potential to carry a scalpel edge, and impressive sturdiness. The shear amount of reducing abuse I've thrown at this little knife over the years and has been in depth. I am not utilizing it to pry open wooden crates, however I have used the ring finish as a screwdriver and to flippantly pry open various things extra instances than I can remember, and it has held up. The good factor is that that same exact quality and design I bought about a decade in the past is retained within the current design.

The SOG Trident was given to me to make use of and to offer my suggestions. The very first thing I seen was it's a bit tougher to open than I expected one handed, because the blade is heavy it doesn't flick open from the tipping point as slick as I would love, but that is not by any means a deterrent. The thumb assist may be little awkward as a result of it's rounded making it laborious if you're carrying gloves or have large fingers. It does have a switch to lock the blade when closed so it does not open by accident because it's spring assisted.

This category describes use around the home, the occasional camping trip, or an occasional carry knife. This is essentially the most versatile class, with the widest range of knives to satisfy the sort of use. You don’t have to fret a lot about issues like weight or dimension, as a result of you'll only carry it as you want it. A knife shouldn’t be complicated however Spyderco Assist Rescue Knife Review (look at these guys) it ought to be well constructed. Don’t buy a gimmicky knife trying to chop corners. You can purchase an awesome knife at an honest price and let it do its job. Save the matches and fishhooks for your Bug out Bag. What number of knives do you need?

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