1. Questions on Normal Subgroups and Factor Groups (11/13) Let G be a group and let H be a subgroup of G. If H is normal in G, then for every a  G and h  H, aha -1 = h. A. TrueB. False If H is normal in G, then for every a  G and h  H, aha -1  H. A. TrueB. False If H is not normal in G, then for every a  G, aH  Ha. A. TrueB. False  V  is normal in D 4. A. TrueB. False  R 180  is normal in D 100. A. TrueB. False If H is abelian, then H is normal in G. A. TrueB. False

2. More Questions We can only form the cosets of H in G if H is normal in G. A. TrueB. False The cosets of H in G only form a group if H is normal in G. A. TrueB. False What is the order of D 100 /  R 180  ? A. 200B. 180C. 100D. 50E. 25 What is the order of (Z 12  U(12)) /  (3, 5)  ? A. 4B. 8C. 12D. 24E. 48 What is the order of 10 in Z ? A. 1B. 10C. 20D.  What is the order of 10 +  15  in Z /  15  A. 1B. 2C. 3D. 10E. 

3. and a few more It makes sense to form the factor group Z n / U(n). A. TrueB. False The factor group Z /  n  is isomorphic to Z n. A. TrueB. False The factor group D 4 /  R 180  is isomorphic to Z 4. A. TrueB. False The factor group S 4 / {(1), (12)(34), (13)(24), (14)(23)} is isomorphic to S 3. A. TrueB. False If H and G / H are abelian, then G must itself be abelian. A. TrueB. False

4. Test #2 Friday Test #2 is on Friday. The format will be the same as Test #1, and in-class portion worth 75 points and then a take-home portion worth 25 points. You may bring a reference sheet. The primary topics are: Chapter 6: Isomorphisms and isomorphic groups Chapter 7: Cosets and Lagrange’s Theorem Chapter 8: External Direct Products Note: Also the Fundamental Theorem of Finite Abelian Groups, which we have stated and used, but not proved. Chapter 9: Normal Subgroups and Factor Groups