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Vierergruppe

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Vierergruppe

The vierergruppe is the Abelian abstract group on four elements that is isomorphic to the finite group C2×C2 and the dihedral group D_2. The multiplication table of one possible representation is illustrated below.

VIV_1V_2V_3
IIV_1V_2V_3
V_1V_1IV_3V_2
V_2V_2V_3IV_1
V_3V_3V_2V_1I

It can be generated by the permutations {1, 2, 3, 4}, {2, 1, 4, 3}, {3, 4, 1, 2}, and {4, 3, 2, 1}.

It has subgroups {I}, {I,V_1}, {I,V_2}, {I,V_3}, and {I,V_1,V_2,V_3} all of which are normal, so it is not a simple group. Each element is in its own conjugacy class.

See also

Dihedral Group D2, Cyclic Group C4, Finite Group C2×C2

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 184-185 and 239-240, 1985.Klein, F. Vorlesungen ueber das Ikosaeder und die Aufloesung der Gleichungen vom fuenften Grade. 1884. Reprinted as Klein, F. Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, 2nd rev. ed. New York: Dover, 1956.

Cite this as:

Weisstein, Eric W. "Vierergruppe." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Vierergruppe.html

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