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Dihedral Group D_5

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The group D_5 is one of the two groups of order 10. Unlike the cyclic group C_(10), D_5 is non-Abelian. The molecule ruthenocene (C_5H_5)_2Ru belongs to the group D_(5h), where the letter h indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248).

D_5 has cycle index given by

Z(D_5)=1/(10)x_1^5+1/2x_2^2x_1+2/5x_5.
DihedralGroupD5Table

Its multiplication table is illustrated above.

The dihedral group D_5 has conjugacy classes {1}, {B,C}, {A,D}, and {E,F,G,H,I}. It has 8 subgroups: {1}, {1,E}, {1,F}, {1,G}, {1,H}, {1,I}, {1,A,B,C,D}, and {1,A,B,C,D,E,F,G,H,I}, of which {1}, {1,A,B,C,D}, and {1, A, B, C, D, E, F, G, H, I} are normal.

See also

Dihedral Group, Dihedral Group D3, Dihedral Group D4

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248, 1985.

Cite this as:

Weisstein, Eric W. "Dihedral Group D_5." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DihedralGroupD5.html

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