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Questions on Normal Subgroups and Factor Groups (11/13)

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. True B. False If H is normal in G , then for every a  G and h  H , aha -1  H.

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Questions on Normal Subgroups and Factor Groups (11/13)

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  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. True B. False • If H is normal in G, then for every a G and h  H, aha-1 H. • A. True B. False • If His not normal in G, then for every a G, aH Ha. • A. True B. False • V is normal in D4. • A. True B. False • R180 is normal in D100. • A. True B. False • If H is abelian, thenH is normal in G. • A. True B. False

  2. More Questions • We can only form the cosets of H in G if H is normal in G. • A. True B. False • The cosets of H in G only form a group if H is normal in G. • A. True B. False • What is the order of D100/ R180 ? • A. 200 B. 180 C. 100 D. 50 E. 25 • What is the order of (Z12  U(12)) / (3, 5) ? • A. 4 B. 8 C. 12 D. 24 E. 48 • What is the order of 10 in Z ? • A. 1 B. 10 C. 20 D.  • What is the order of 10 + 15 in Z / 15 • A. 1 B. 2 C. 3 D. 10 E. 

  3. and a few more • It makes sense to form the factor group Zn / U(n). • A. True B. False • The factor group Z / n is isomorphic to Zn. • A. True B. False • The factor group D4 / R180 is isomorphic to Z4. • A. True B. False • The factor group S4 / {(1), (12)(34), (13)(24), (14)(23)}is isomorphic to S3. • A. True B. False • If H and G / H are abelian, then G must itself be abelian. • A. True B. 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

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