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Normal Subgroups and Factor Groups (11/11)

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.

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Normal Subgroups and Factor Groups (11/11)

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  1. Normal Subgroups and Factor Groups (11/11) • Definition. A subgroup H of a group G is called normalif for every a G, the left cosetaH is the same set as the right cosetHa. 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). • Anis normal in Sn. • 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 {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 Siandany b from Sjand add a + b. It must be in some Sk. Then we defineSi+ Sjto be Sk. • Definition. This induced operation is called well-defined if we always get the same answer Skno matter what elements we pick from SiandSj.

  3. 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 isS1+ 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?

  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 ah1 be any element of aHand 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.

  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: • 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?

  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.

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