Editing Group (mathematics) (section)

=== Cyclic groups ===
{{Main|Cyclic group}}
[[Image:Cyclic group.svg|right|thumb|upright|The 6th complex roots of unity form a cyclic group. <math>z</math> is a primitive element, but <math>z^2</math> is not, because the odd powers of <math>z</math> are not a power of {{tmath|1= z^2 }}.|alt=A hexagon whose corners are located regularly on a circle]]
A ''cyclic group'' is a group all of whose elements are [[power (mathematics)|powers]] of a particular element {{tmath|1= a }}.{{sfn|Lang|2005|loc=§II.1|p=22}} In multiplicative notation, the elements of the group are
<math display=block>\dots, a^{-3}, a^{-2}, a^{-1}, a^0, a, a^2, a^3, \dots,</math>
where <math>a^2</math> means {{tmath|1= a\cdot a }}, <math>a^{-3}</math> stands for {{tmath|1= a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1} }}, etc.{{efn|The additive notation for elements of a cyclic group would be <!--use {{math}}, since <math> in footnotes is unreadable on mobile devices-->{{math|''t'' ⋅ ''a''}}, where {{math|''t''}} is in {{math|'''Z'''}}.}} Such an element <math>a</math> is called a generator or a [[Primitive root modulo n|primitive element]] of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as
<math display=block>\dots, (-a)+(-a), -a, 0, a, a+a, \dots.</math>

In the groups <math>(\Z/n\Z,+)</math> introduced above, the element <math>1</math> is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are {{tmath|1= 1 }}. Any cyclic group with <math>n</math> elements is isomorphic to this group. A second example for cyclic groups is the group of {{tmath|1= n }}th [[root of unity|complex roots of unity]], given by [[complex number]]s <math>z</math> satisfying {{tmath|1= z^n=1 }}. These numbers can be visualized as the [[vertex (graph theory)|vertices]] on a regular <math>n</math>-gon, as shown in blue in the image for {{tmath|1= n=6 }}. The group operation is multiplication of complex numbers. In the picture, multiplying with <math>z</math> corresponds to a [[clockwise|counter-clockwise]] rotation by 60°.{{sfn|Lang|2005|loc=§II.2|p=26}} From [[field theory (mathematics)|field theory]], the group <math>\mathbb F_p^\times</math> is cyclic for prime <math>p</math>: for example, if {{tmath|1= p=5 }}, <math>3</math> is a generator since {{tmath|1= 3^1=3 }}, {{tmath|1= 3^2=9\equiv 4 }}, {{tmath|1= 3^3\equiv 2 }}, and {{tmath|1= 3^4\equiv 1 }}.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element {{tmath|1= a }}, all the powers of <math>a</math> are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to {{tmath|1= (\Z, +) }}, the group of integers under addition introduced above.{{sfn|Lang|2005|p=22|loc=§II.1 (example 11)}} As these two prototypes are both abelian, so are all cyclic groups.

The study of finitely generated abelian groups is quite mature, including the [[fundamental theorem of finitely generated abelian groups]]; and reflecting this state of affairs, many group-related notions, such as [[Center (group theory)|center]] and [[commutator]], describe the extent to which a given group is not abelian.{{sfn|Lang|2002|loc=§I.5|pp=26, 29}}