Let G be a subgroup of some dihedral group. For each x in G, define phi(x) = { (+1 if x is a rotation), (-1 if x is a reflection) }.
How do you prove that phi is a homomorphism from G to the multiplicative group {+1, -1). Also, what is the kernal of phi????
2007-12-01 14:34:56 · 3 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
The kernel would be whatever is mapped to the identity element, and that's spelled out explicitly in the definition of the mapping! (I.e., it's the rotations).
To prove that it's a homomorphism, you basically have to prove that the image of the identity is the identity (easy) and that the image of a product is the product of the images.
Well, what's the product of two rotations? Two reflections? A rotation and a reflection?
2007-12-01 18:33:31 · answer #1 · answered by Curt Monash 7 · 0⤊ 0⤋
i'm no longer particular what R* is. probable R - {0}? phi is nicely defined because that for each x phi(x) is unique Phi(ab) = |ab| =|a||b| = phi(a)phi(b). So phi is a homomorphism. If phi(x) = a million then |x| = a million and x = + or - a million, the kernel of phi.
2016-10-25 07:06:14 · answer #2 · answered by leisure 4 · 0⤊ 0⤋
Kernal?
2007-12-01 14:37:39 · answer #3 · answered by Lelar 6 · 0⤊ 3⤋