dihedral group (idea) by pitcher

COMMUTABILITY OF ELEMENTS IN D_2n

If n=2k is even and n>=4, then r^k is an element of order 2 which commutes with all elements of D_2n. Also, r^k is the only nonidentity element of D_2n which commutes with all elements of D_2n.

PROOF:

r^k is an element of order 2 which commutes with all elements of D_2n.
rⁿ=1 --> r^2k=1
Therefore r^k is an element of order 2. |r^k|=2

To show commutability, break into two cases:
- let rⁱ be an element in D_2n. Then (r^k)(rⁱ) = r^k+i = r^i+k = (rⁱ)(r^k)
- Let rⁱs be an element in D_2n. Then (rⁱs)(r^k) = rⁱ(sr^ks)s = rⁱr^-ks = r^k(rⁱs).

Suppose some element x of D_2n commutes with every other element.

suppose x=rⁱ. Then xs=sx => rⁱs = srⁱ = r^-i => rⁱ = r^-i => r²ⁱ=1, so i=k.

Now suppose n is odd. Argue as above with x=srⁱ and x=rⁱ. If x=srⁱ, then r²=1 which is not possible. If x=rⁱ we get r²ⁱ=1 which is also not possible as then either n|2i or n|i as n is odd, so x=1. Therefore, if n is odd, only the identity commutes with all elements in D_2n

groups of order 8	Sylow's theorem	Leonardo's theorem	Generators and relations for groups
representations of groups	crystallographic groups	symmetry group	Torsion angle
Má Vlast	symmetric group	tetrahedral group	HTML symbol reference
classification theorem of finite simple groups	Point Group	Example of a normal subgroup of a normal subgroup which is not normal	midpoint
Symmetry	Generator	Dihedral	Geometric shapes
permutation	reflection	rotation