Some groups exhibit, in miniature, the structure of other groups. A group H contains a ‘miniature version’ of a group G when there is a function of G to H, usually a many-to-one function, which preserves the group structure of G. Such a function is called a homomorphism. What we mean when we say that a function ‘preserves the group structure’ is that if g1 ↦ h1 and g2 ↦ h2, then g1g2 ↦ h1h2. In ordinary language we describe this by saying that the image of a product is the product of the images.

Concurrent reading: Fraleigh, sections 11 and 12; Green, chapter 7.

1. 1 If A and B are matrices in GL(2, F) and A and B belong to the same right coset of SL(2, F), what can you say about det A and det B?

2. 2 If A and B are matrices in GL(2, F) and det A = det B, does it follow that A and B belong to the same right coset of SL(2, F)?

3. 3 What is the index of SL(2, ℤ3) in GL(2, ℤ3) and what is the order of SL(2, ℤ3)? (See qn 13.32)

4. 4 What is the index of SL(2, ℤ5) in GL(2, ℤ5)? What is the order of SL(2, ℤ5)?

5. 5 If A and B are matrices in GL(2, F) and det A = det B, does it follow that A and B belong to the same left coset of SL(2, F)?

6. 6 […]