- #1
hsong9
- 80
- 1
Homework Statement
Let Dn = {1,a,..an-1, b, ba,...ban-1} with |a|=n, |b|=2,
and aba = b.
show that every subgroup K of <a> is normal in Dn.
The Attempt at a Solution
First, we show <a> is normal in Dn.
<a> = {1,a,...an-1} has index 2 in Dn and so is normal
by Thm (If H is a subgroup of index 2 in G, then H is normal in G.)
Next, Since <a> is cyclic, K is also cyclic and abelian.
Let k in K, x in G and 1 in G. ( G = Dn)
k = 1k = (xx-1)k = x(kx-1) because K is abelian.
k in K => xkx-1 in K for all x in G
=> K is a normal in G = Dn.