Order of elements

in general, no. it may be that |ab| isn't even finite.

for example, if G is the quotient of the free group on two generators defined by:

a² = b² = e

then ab is not of finite order, even though both a and b are of order 2.

 

for the dihedral group D_n, the elements of order 2 depend on whether or not n is even.

if n is odd, then the sylow 2-subgroups are all of order 2, because half the elements are of odd order (the rotations). geometrically, it's obvious the reflections are of order 2, but we can prove this algebraically, as well:

given: rⁿ = 1, s² = 1, and rs = sr^-1, let's calculate (sr^k)².

first, a small detour:

lemma: rs = sr^-1 → r^ks = sr^-k.


for k = 1, this is given by the defining relations on D_n.

suppose this is true for k = m.

then r^m+1s = r(r^ms) = r(sr^-m) = (rs)r^-m
= (sr^-1)r^-m = sr^-(m+1),

so the general result holds by induction on k.

corollary: sr^k = r^-ks

since r^-k = r^n-k,

we have r^n-ks = sr^k-n, so rⁿr^-ks = sr^kr^-n, so the result follows, because both rⁿ and r^-n are the identity.

ok, back to showing what the square of a reflection is:

(sr^k)² = (sr^k)(sr^k)

= (r^-ks)(sr^k) = r^-k(ss)r^k

= r^-kr^k = 1.

so every reflection has order 2. now, since 9 is odd, this means we have 9 sylow 2-subgroups, one for each reflection.

(if n is even, we have a slightly more involved situation, because now 4 divides the order of the group).