- #1

- 16

- 0

Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- #1

- 16

- 0

Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?

- #2

Science Advisor

Homework Helper

- 9,426

- 6

This is either a consequence of the structure theorem for abelian groups, or an explanation of that theorem, depending on how you look at it.

We could look a bit deeper: suppose that G and H are cyclic, when is GxH cyclic? Well, suppose that we claim (g,h) is a cyclic generator of GxH. Well, g must be a generator of G with order p, say, and h a generator of H with order q, and (g,h) must have order pq. If I raise (g,h) to the power p, then it is (1,k) for some k in H, and in order for (g,h) to have order pq k must have order q, i.e. also be a generator for H.

Pulling out the important thing: H must have an element h of order so that h^p still has order q. But this is if and only if p is coprime to q (this is an elementary fact you may have learned already).

Share:

- Replies
- 1

- Views
- 409

- Replies
- 7

- Views
- 840

- Replies
- 3

- Views
- 164

- Replies
- 11

- Views
- 549

- Replies
- 1

- Views
- 534

- Replies
- 2

- Views
- 176

- Replies
- 0

- Views
- 529

- Replies
- 7

- Views
- 905

- Replies
- 12

- Views
- 2K

- Replies
- 4

- Views
- 1K