Let G be a group. Define j: G

1

• Let G be a group. Define j G G by j(x) x.
• The First Isomorphism Theorem says
• 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 j G e by j(x) e.
  
• The First Isomorphism Theorem says
• 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
  Q8/ltIgt ? V, we first
• (a) Define j Q8 ltIgt
• Define j Q8 Q8/ltIgt
• Define j Q8/ltIgt V
• (d) Define j Q8 V
• (e) Show Q8/ltIgt and V are both abelian.
• (f) Show Q8/ltIgt and V are both cyclic.

4

• To use the First Isomorphism Theorem to show that
  Q8/ltIgt ? V, we first define j Q8 V. We then
  
• (a) Show j is a homomorphism.
• Show j is an isomorphism.
• (c) Show Q8/ltIgt and V are both abelian.
• (d) Show Q8/ltIgt and V are both cyclic.

5

• To use the First Isomorphism Theorem to show that
  Q8/ltIgt ? V, we first define j Q8 V. We then
  
• Show j is onto.
• (b) Show j is one-to-one
• (c) Show j is an isomorphism.
• (c) Show Q8/ltIgt and V are both abelian.
• (d) Show Q8/ltIgt and V are both cyclic.

6

• To use the First Isomorphism Theorem to show that
  Q8/ltIgt ? V, we first define j Q8 V. We then
  
• (a) Show j is one-to-one
• Show j is an isomorphism.
• (c) Show ltIgt and V are isomorphic.
• (d) Show ker j ltIgt
• (e) Show Q8/ltIgt and V are both abelian.
• (f) Show Q8/ltIgt and V are both cyclic.

7

• To start the proof, we
• (a) Define j G G/K
• Define j G/H K/H
• Define j G/H G/K
• (d) Define j G/K G
• (e) Show j is a homomorphism.

8

• The thing that goes j(here) is
• (a) g (b) x
• Kg (d) Hg
• (g, k) (f) (h, k)
• (g) hg (h) kg

9

• The thing that goes j(Hg) here is
• (a) g (b) x
• Kg (d) Hg
• (g, k) (f) (h, k)
• (g) hg (h) kg

10

• We then
• (a) Show j is a homomorphism.
• Show j 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
• Show j is one-to-one.
• (b) Show j is onto.
• (c) Show j is an isomorphism.
• (c) Show Q8/ltIgt and V are both abelian.
• (d) Show Q8/ltIgt and V are both cyclic.

12

• We then
• (a) Show j is one-to-one
• Show j is an isomorphism.
• (c) Show ker j G/K.
• (d) Show ker j G/H
• Show ker j K/H
• (f) Show Q8/ltIgt and V are both cyclic.