Any help please Find groups G and H and a surjective homomorphism ?: G rightarrow H such
that Z(G) Z(H)
Solution
When is the abelianization map, you get lots of examples. For example:
Let G be the group of quaternions: {±1, ±i, ±j, ±k}
Let be the map: :G--->G/{±1} = H
{±1} is the commutator subgroup of G, and so the result is the abelianization of G, which is
isomorphic to Z2 x Z2.
At any rate, H is itself Abelian, so Z(H)=H. However, Z(G)={±1}, whose image under is just
the identity of H. Therefore Z(G) is NOT equal to Z(H)..
Science 7 - LAND and SEA BREEZE and its Characteristics
Any help please Find groups G and H and a surjective homomorphism .pdf
1. Any help please Find groups G and H and a surjective homomorphism ?: G rightarrow H such
that Z(G) Z(H)
Solution
When is the abelianization map, you get lots of examples. For example:
Let G be the group of quaternions: {±1, ±i, ±j, ±k}
Let be the map: :G--->G/{±1} = H
{±1} is the commutator subgroup of G, and so the result is the abelianization of G, which is
isomorphic to Z2 x Z2.
At any rate, H is itself Abelian, so Z(H)=H. However, Z(G)={±1}, whose image under is just
the identity of H. Therefore Z(G) is NOT equal to Z(H).
