|
|
| Line 1: |
Line 1: |
| {{Group theory sidebar}}
| | I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Office supervising is where my main earnings comes from but I've usually wanted my own company. The favorite pastime for him and his children is style and he'll be beginning something else alongside with it. Mississippi is where his home is.<br><br>my blog post - phone psychic readings ([http://brazil.amor-amore.com/irboothe brazil.amor-amore.com]) |
| | |
| In [[mathematics]], specifically in [[abstract algebra]], a '''torsion-free abelian group''' is an [[abelian group]] which has no non-trivial [[torsion (algebra)|torsion]] elements; that is, a [[group (mathematics)|group]] in which the [[group operation]] is [[commutative property|commutative]] and the [[identity element]] is the only element with finite [[order (group theory)|order]]. That is, multiples of any element other than the identity element generate an infinite number of distinct elements of the group.
| |
| | |
| == Definitions ==
| |
| {{main|Abelian group}}
| |
| | |
| An '''abelian group''' <math> \langle G, * \rangle </math> is a [[set (mathematics)|set]] ''G'', together with a [[binary operation]] * on ''G'', such that the following axioms are satisfied:
| |
| | |
| ;Associativity: For all ''a'', ''b'' and ''c'' in ''G'', (''a'' * ''b'') *''c'' = ''a'' * (''b'' * ''c'').
| |
| ;Identity element: There is an element ''e'' in ''G'', such that ''e'' * ''x'' = ''x'' * ''e'' = ''x'' for all ''x'' in ''G''. This element ''e'' is an '''identity element''' for * on ''G''.
| |
| ;Inverse element: For each ''a'' in ''G'', there is an element ''a''′ in ''G'' with the property that ''a''′ * ''a'' = ''a'' * ''a''′ = ''e''. The element ''a''′ is an '''inverse of''' ''a'' '''with respect to *'''.
| |
| ;Commutativity: For all ''a'', ''b'' in ''G'', ''a'' * ''b'' = ''b'' * ''a''.<ref>{{harvtxt|Fraleigh|1976|pp=18−20}}</ref><ref>{{harvtxt|Herstein|1964|pp=26−27}}</ref><ref>{{harvtxt|McCoy|1968|pp=143−146}}</ref>
| |
| | |
| | |
| {{main|Order (group theory)}}
| |
| ;Order: For this definition, note that in an abelian group, the binary operation is usually called addition, the symbol for addition is “+”<ref>{{harvtxt|Fraleigh|1976|p=27}}</ref> and a repeated sum, <math>a + a + \cdots + a</math> of the same element appearing ''n'' times is usually abbreviated “''na''”.<ref>{{harvtxt|Fraleigh|1976|p=30}}</ref> Let ''G'' be a group and ''a'' ∈ ''G''. If there is a positive integer ''n'' such that ''na'' = ''e'', the least such positive integer ''n'' is the '''order of''' ''a''. If no such ''n'' exists, then ''a'' is of '''infinite order'''.<ref>{{harvtxt|Fraleigh|1976|pp=50,72}}</ref><ref>{{harvtxt|Herstein|1964|p=37}}</ref><ref>{{harvtxt|McCoy|1968|p=166}}</ref>
| |
| | |
| | |
| {{main|Torsion (algebra)}}
| |
| ;Torsion: A group ''G'' is a '''torsion group''' if every element in ''G'' is of finite order. ''G'' is '''torsion free''' if no element other than the identity is of finite order.<ref>{{harvtxt|Fraleigh|1976|p=78}}</ref><ref>{{harvtxt|Lang|2002|p=42}}</ref><ref>{{harvtxt|Hungerford|1974|p=78}}</ref>
| |
| | |
| ==Properties==
| |
| * A torsion-free abelian group has no non-trivial finite [[subgroup]]s.
| |
| * A [[generating set of a group#finitely generated group|finitely generated]] torsion-free abelian group is [[free abelian group|free]].<ref>{{harvtxt|Lang|2002|p=45}}</ref>
| |
| | |
| ==See also==
| |
| *[[Rank of an abelian group]]
| |
| *[[Torsion-free abelian groups of rank 1]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
| |
| | |
| * {{ citation | first1 = I. N. | last1 = Herstein | year = 1964 | isbn = 978-1114541016 | title = Topics In Algebra | publisher = [[Blaisdell Publishing Company]] | location = Waltham }}
| |
| | |
| * {{ citation | first1 = Thomas W. | last1 = Hungerford | year = 1974 | isbn = 0-387-90518-9 | title = Algebra | publisher = [[Springer-Verlag]] | location = New York }}.
| |
| | |
| * {{ citation | first1 = Serge | last1 = Lang | year = 2002 | isbn = 0-387-95385-X | title = Algebra | edition = Revised 3rd | publisher = [[Springer-Verlag]] | location = New York }}.
| |
| | |
| * {{ citation | first1 = Neal H. | last1 = McCoy | year = 1968 | title = Introduction To Modern Algebra, Revised Edition | publisher = [[Allyn and Bacon]] | location = Boston | lccn = 68-15225 }}
| |
| | |
| {{Group navbox}}
| |
| | |
| {{DEFAULTSORT:Torsion-free Abelian Group}}
| |
| [[Category:Algebraic structures]]
| |
| [[Category:Group theory]]
| |
| [[Category:Abelian group theory]]
| |
| [[Category:Properties of groups]]
| |
I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Office supervising is where my main earnings comes from but I've usually wanted my own company. The favorite pastime for him and his children is style and he'll be beginning something else alongside with it. Mississippi is where his home is.
my blog post - phone psychic readings (brazil.amor-amore.com)