Can someone please help me with these exercises as I think I'm getting them wrong. Thanks you for your help.

Consider the dihedral group

D4, and let and be defined in the usual way. Show that = −1, and hence that k = −k. Thus show that k has order 2 for any k.

Let

G = {a, b, c, d, e, f, g, h}, and let *be a group operation on G. A partial Cayley table for *is attached.

1.Complete the Cayley table above [hint: the operation must be associative

2. Write down <

a>, <b>, <c>, <d>, <e>, <f>, <g>, <h>, <b,e>, <c,e>

3. Write the elements of G as products of two fixed generating elements (you may use

1 to denote the identity).

4.Show that (G,*) is not isomorphic to D4 [Hint: question 1, order of elements] or

to Z/4Z × Z/2Z.

5. Show that <b>is normal, and write out the left cosets of this subgroup. Then give

the group table for G/<b>.

6. Give an isomorphism from G/<b>to either Z/4Z or Z/2Z × Z/2Z. In the othercase, explain why no isomorphism exists.