In: Advanced Math
Let G = Z x Z and H = {(a, b) in Z x Z | 8 divides a+b}
a. Prove directly that H is a normal subgroup in G (use the fact
that closed under composition and inverses)
b. Prove that G/H is isomorphic to Z8.
c. What is the index of [G : H]?
Recall some Basic facts:
For any integer
SOLUTION:
is an abelian group.
such that
Claim:
is a Normal subgroup of
.
Since
is abelian group this implies every subgroup of
must be normal subgroup. So to prove
is normal it is enough to show that
is a subgroup of
.
So from the above discussion we conclude that
is a subgroup of
and since
is abelian so
is normal in
.
Solution of (b) & (c)
NOTE: for an arbitrary
we get a preimage.so this is true for all
. Hence
is onto.
Remark: