1. Normal Subgroups and Factor Groups (11/11) Definition. A subgroup H of a group G is called normal if for every a  G, the left coset aH is the same set as the right coset Ha. If this holds, we write H  G. As you observed in the recent hand-in, aH = Ha is equivalent to aHa -1 = H, so this corresponds to the definition we gave on Test 1. Examples: Every subgroup of every abelian group is normal! Z(G) is normal in G. (E.g.,  R 180  is normal in D n, n even.) SL(2, R) is normal in GL(2, R). A n is normal in S n. In fact, if [G:H] = 2, H is normal in G. (Why??)

2. Induced, well-defined operations Suppose S is a set with a binary operation + and suppose {S 1, S 2,..., S n } is a partition of S into subsets. We (try to) induce an operation on this collection of sets as follows: What is S i + S j ? Well, take any a from S i and any b from S j and add a + b. It must be in some S k. Then we define S i + S j to be S k. Definition. This induced operation is called well-defined if we always get the same answer S k no matter what elements we pick from S i and S j.

3. Example of well-definedness (and non-) Consider Z 10 under its normal addition mod 10: Suppose we partition Z 10 as follows: S 1 = {0, 8}, S 2 = {1, 7}, S 3 = {2, 4}, S 4 = {5, 9}, S 5 = {3, 6}. Inducing the operation on these sets, what is S 1 + S 2 ? Well if we pick 0 from S 1 and 7 from S 2, we get S 2 as our answer, but if we pick 8 and 7 we get S 4. Clearly this partition does not allow a well-defined induced operation. Now suppose we partition Z 10 as follows: S 1 = {0, 5}, S 2 = {1, 6}, S 3 = {2, 7}, S 4 = {3, 8}, S 5 = {4, 9}. Try some examples now. Here we do have a well-defined induced operation. What are these sets in this second partition?

4. Factor Groups Definition. Let G be a group and let H be a normal subgroup of G. The factor group G / H is the set of left (or right) cosets of H under the operation induced by G’s operation, that is, for all a and b in G, (aH)(bH) = abH. This operation is called coset multiplication. Theorem. Coset multiplication is well-defined provided that H is normal in G. Proof. Let ah 1 be any element of aH and let bh 2 be any element of bH. Then since H is normal, we know that Hb = bH, so h 1 b = bh 3 for some h 3 in H. But now we have: (ah 1 )(bh 2 ) = a(h 1 b)h 2 = a(bh 3 )h 2 = (ab)(h 3 h 2 )  abH.

5. Is G / H a group? And examples Coset multiplication is a well-defined binary operation which inherits associativity from G’s operation. Check. What is the identity coset? Check. What is the inverse coset of aH ? Check. Note that |G / H| = [G: H] (simply the number of cosets) Examples: Z 10 /  5  was written down two slides above. What group is Z / 4Z isomorphic to? What group is D 4 /  R 90  isomorphic to? What group is D 4 /  R 180  isomorphic to? What group is R* / {1,-1} isomorphic to?

6. Assignment for Wednesday Read Chapter 9 to page 193 and do Exercises 1, 2, 6, 7, 12, 13, 14, 18, 19 on pages 200-201. No new material on Wednesday! Test #2 on Friday.