is the unique group of group order 2. is both Abelian and cyclic. Examples include the point
, and , the integers modulo 2 under addition ( ), and the modulo
multiplication groups , , and (which are the only modulo multiplication groups isomorphic
is also trivially simple, and forms the subject for
the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern
University mathematics department a capella group "The Klein Four."
cycle graph is shown above, and the cycle
where 1 is the identity element.
multiplication table is illustrated above
and enumerated below.
conjugacy classes are and . The only subgroups of are the trivial group and entire group , both of which are trivially normal.
The irreducible representation for the
group is .
See also Cyclic Group
Cyclic Group C3
Cyclic Group C4
Cyclic Group C6
Cyclic Group C8
Cyclic Group C10
Cyclic Group C12
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References The Klein Four. "Finite Simple Group (of Order Two)."
http://www.math.northwestern.edu/~matt/kleinfour/. Cite this as:
Weisstein, Eric W. "Cyclic Group C_2."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/CyclicGroupC2.html