The Kernel takes an element from the first group and maps it to the identity in the second group.
Let and be groups with identities and , respectively. Let be a homomorphism of groups. Let be the kernel of . Let . Which of the following is/are necessarily true?
a)
b)
c)
d)
e)
I've never taken abstract algebra before so I have never seen anything like this. Any help would be greatly appreciated.
by definition, . so clearly (c) is always true.
(a) is sometimes true, but not always. for example, let and . the map:
given by is a homomorphism, but and ,
but clearly .
(b) is NEVER true. the kernel of f lies entirely in G_{1} (unless G_{1} is a subgroup of G_{2}, in which case ).
(d) is likewise NEVER true, f(x) is always in G_{2} (but see above).
(e) if G_{1}, G_{2} are distinct groups, isn't even defined.