1. Let G be a group. Define  : G ® G by  ( x ) = x. The First Isomorphism Theorem says: (a) j(ab) = j(a)j(b)(b) j is onto. (c) G  G(d) G  {e} (e) G/{e}  G(f) G/G  {e}

2. Let G be a group. Define  : G ® {e} by  ( x ) = e. The First Isomorphism Theorem says: (a) j(ab) = j(a)j(b)(b) j is onto. (c) G  G(d) G  {e} (e) G/{e}  G(f) G/G  {e}

3. To use the First Isomorphism Theorem to show that Q 8 /  V, we first : (a) Define  : Q 8  (b) Define  : Q 8  Q 8 / (c) Define  : Q 8 /  V (d) Define  : Q 8  V (e) Show Q 8 / and V are both abelian. (f) Show Q 8 / and V are both cyclic.

4. To use the First Isomorphism Theorem to show that Q 8 /  V, we first d efine  : Q 8  V. We then: (a) Show  is a homomorphism  (b) Show  is an isomorphism. (c) Show Q 8 / and V are both abelian. (d) Show Q8/ and V are both cyclic.

5. To use the First Isomorphism Theorem to show that Q 8 /  V, we first d efine  : Q 8  V. We then: (a) Show  is onto  (b) Show  is one-to-one (c) Show  is an isomorphism. (c) Show Q 8 / and V are both abelian. (d) Show Q8/ and V are both cyclic.

6. To use the First Isomorphism Theorem to show that Q 8 /  V, we first d efine  : Q 8  V. We then: (a) Show  is one-to-one (b) Show  is an isomorphism. (c) Show and V are isomorphic. (d) Show ker  (e) Show Q 8 / and V are both abelian. (f) Show Q 8 / and V are both cyclic.

7. To start the proof, we: (a) Define  : G  G/K (b) Define  : G/H  K/H (c) Define  : G/H  G/K (d) Define  : G/K  G (e) Show  is a homomorphism.

8. The thing that goes  ( here ) is: (a) g(b) x (c) Kg(d) Hg (e) (g, k) (f) (h, k) (g) hg(h) kg

9. The thing that goes  (Hg) = here is: (a) g(b) x (c) Kg(d) Hg (e) (g, k) (f) (h, k) (g) hg(h) kg

10. We then: (a) Show  is a homomorphism  (b) Show  is an isomorphism. (c) Show G/H and G/K are both abelian. (d) Show G/H and G/K are both cyclic.

11. We then: (a) Show  is one-to-one  (b) Show  is onto. (c) Show  is an isomorphism. (c) Show Q 8 / and V are both abelian. (d) Show Q8/ and V are both cyclic.

12. We then: (a) Show  is one-to-one (b) Show  is an isomorphism. (c) Show ker  = G/K. (d) Show ker  G/H (e)Show ker  K/H (f) Show Q 8 / and V are both cyclic.