rev1 - 7 Group actions 8 Subgroups cosets and Lagrange’s...

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Math 4124 Wednesday, February 16 First Test Review The test will cover up to (and including) section 3.2. Topics will include 1. The order of an element. If n and r are positive integers and | g | = n , then | g r | = n / ( n , r ) . 2. Direct product of two groups (as in problem 1 of sample test). 3. Dihedral and quaternion groups. 4. Symmetric groups, the order of an element in S n . 5. Cyclic groups, subgroups of a cyclic group. If n is a positive integer, then a cyclic group of order n is isomorphic to Z / n Z . Furthermore for each positive integer d ± ± n , it has a unique subgroup of order d , namely h n / d i . 6. Properties of isomorphic groups.
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Unformatted text preview: 7. Group actions. 8. Subgroups, cosets and Lagrange’s theorem. 9. Centralizers, center and normalizers. 10. Homomorphisms and normal subgroups. All subgroups of an abelian group are nor-mal. 11. If Z is a central subgroup of G (so Z ≤ Z ( G ) ) and G / Z is cyclic, then G is abelian. 12. A group of prime order p is isomorphic to Z / p Z . Test on Monday, February 21. One problem will be identical to one of the ungraded homework problems. Review session on Sunday, February 20 at 4:45 p.m. in McBryde 210....
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