Dihedral Group D_5
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. |
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 D4Explore with Wolfram|Alpha
More things to try:
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248, 1985.Referenced on Wolfram|Alpha
Dihedral Group D_5Cite this as:
Weisstein, Eric W. "Dihedral Group D_5." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DihedralGroupD5.html