Question

In: Advanced Math

Let f: Q8 to D8 be a homomorphism from the quaternion Q8 to the dihedral group...

Let f: Q8 to D8 be a homomorphism from the quaternion Q8 to the dihedral group D8 of order 8. Suppose that f (i) = rs and f( j) = r^2.

(a) Find f (k) and f(-k) (expressed in standard form r^i*s^i for suitable i and j) ·

(b) List the elements in the kernel of f in standard form.

(c) The quotient group Q8/ker(f) is isomorphic to which one of the groups C2, C4 or C2xC2 why?

Expert Solution

Problem (a)

Problem (b)

As we have seen above,

So, are not in the kernel.

Now,

And

So, are also not in the kernel.

Thus, the only elements in the kernel are

Problem (c)

Now, if you quotient by , that means is same as

So, and

And also,

So, it is an abelian group of order , namely where

Therefore,

Colby Messinger answered 2 years ago

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