Question

In: Advanced Math

Let G = Z x Z and H = {(a, b) in Z x Z |...

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]?

Solutions

Expert Solution

Recall some Basic facts:

For any integer

divides that is .
If , then .
If and , then for arbitrary integers and

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:

Colby Messinger answered 1 year ago

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