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., R180 is normal in Dn, n even.)• SL(2, R) is normal in GL(2, R).• An is normal in Sn.• In fact, if [G:H] = 2, H is normal in G. (Why??)

Induced, well-defined operations• Suppose S is a set with a binary operation + and suppose

{S1, S2,..., Sn } is a partition of S into subsets. We (try to) induce an operation on this collection of sets as follows: What is Si + Sj ? Well, take any a from Si and any b from Sj and add a + b. It must be in some Sk. Then we define Si + Sj to be Sk.

• Definition. This induced operation is called well-defined if we always get the same answer Sk no matter what elements we pick from Si and Sj .

Example of well-definedness (and non-)

• Consider Z10 under its normal addition mod 10:• Suppose we partition Z10 as follows:

S1= {0, 8}, S2= {1, 7}, S3= {2, 4}, S4= {5, 9}, S5= {3, 6}.• Inducing the operation on these sets, what is S1 + S2?

Well if we pick 0 from S1 and 7 from S2, we get S2 as our answer, but if we pick 8 and 7 we get S4. Clearly this partition does not allow a well-defined induced operation.

• Now suppose we partition Z10 as follows:S1= {0, 5}, S2= {1, 6}, S3= {2, 7}, S4= {3, 8}, S5= {4, 9}. Try some examples now. Here we do have a well-defined induced operation.

• What are these sets in this second partition?

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 ah1 be any element of aH and let bh2 be any element of bH. Then since H is normal, we know that Hb = bH, so h1b = bh3 for some h3 in H. But now we have: (ah1)(bh2) = a(h1b)h2 = a(bh3)h2 = (ab)(h3h2) abH.

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: • Z10 / 5 was written down two slides above.• What group is Z / 4Z isomorphic to?• What group is D4 / R90 isomorphic to?• What group is D4 / R180 isomorphic to?• What group is R* / {1,-1} isomorphic to?

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.