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Group theory

Group theory. C 2 1 C 2 m 1 m 1 1 C 2 1 1 m C 2 C 2 1 m m 1. Isomorphism: Two groups isomorphic if they have same type of multiplication table.

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Group theory

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  1. Group theory C2 1 C2 m 1 m 1 1 C2 1 1 m C2 C2 1 m m 1 Isomorphism: Two groups isomorphic if they have same type of multiplication table

  2. Group theory 222 1 C2 C2 C2 mm2 1 m m' C2 1 1 C2 C2 C2 1 1 m m' C2 C2 C2 1 C2 C2 m m 1 C2 m' C2 C2 C2 1 C2 m' m' C2 1 m C2 C2 C2 C2 1 C2 C2 m' m 1 Isomorphism: Two groups isomorphic if they have same type of multiplication table

  3. Group theory 2 3 222 1 C2 C2 C2 4 1 C4 C4 C4 1 1 C2 C2 C2 1 1 C4 C4 C4 C2 C2 1 C2 C2 C4 C4 C4 C4 1 C2 C2 C2 1 C2 C4 C4 C4 1 C4 C2 C2 C2 C2 1 C4 C4 1 C4 C4 2 3 2 3 2 2 3 3 2 3 Isomorphism: Two groups isomorphic if they have same type of multiplication table

  4. Point Symmetry n n n 3 2 Rotation axes 1 A A2 A3 … Rotoinversion axes Aa i (Aa i)2 (Aa i)3 …… Aa i ( = 1) Aa i Aa Aa i …… Aa ( = 1) Order of improper cyclical rotation group always even

  5. Point Symmetry n n n 3 2 3 2 4 5 6 2 2 3 Rotation axes 1 A A2 A3 … Rotoinversion axes Aa i (Aa i)2 (Aa i)3 …… Aa i ( = 1) Aa i Aa Aa i …… Aa ( = 1) Suppose n odd (3) Aa i Aa Aa i Aa Aa i Aa ( = 1) Aa i Aa i Aa Aa i Aa ( = 1)

  6. Point Symmetry n n n 3 2 3 5 6 4 2 2 2 3 2 2 Rotation axes 1 A A2 A3 … Rotoinversion axes Aa i (Aa i)2 (Aa i)3 …… Aa i ( = 1) Aa i Aa Aa i …… Aa ( = 1) Suppose n odd (3) Aa i Aa Aa i Aa Aa i Aa ( = 1) Aa i Aa i Aa Aa i Aa ( = 1) 1 Aa Aa g = 3 h = i i Aa i Aa i g h = 3

  7. Point Symmetry Rotoinversion axes 1 Aa Aa g = 3 h = i i Aa i Aa i g h = 3 2 2 +

  8. Point Symmetry Suppose n even, n/2 odd (2 or 6) Aa i Aa Aa i Aa Aa i Aa ( = 1) Aa Aπi Aa Aπi Aa Aa Aπi 1 1 Aa Aa m Aa mAa m 3 2 4 5 6 2 4 4 2 2 4 4 2 +

  9. Point Symmetry Suppose n even, n/2 even (4) Aa i Aa Aa i Aa ( = 1) (That's all) 3 2 4

  10. Point Symmetry Subgroups: G g 1 1 2 1 2 3 1 3 4 1 2 4 6 1 2 3 6

  11. Point Symmetry Subgroups: G g 1 1 1 2 1 2 3 1 3 1 3 4 1 2 1 4 6 1 3 2 6

  12. Point Symmetry Group combinations: {C2} {C2'} ({222}) 1 C2 1 1 C2 C2' C2' C2 C2' (= C2")

  13. Point Symmetry Group combinations: {C2} {C2'} ({32}) 1 C3 C3 1 1 C3 C3 C2 C2 C2 C3 C2 C3 C2 C2'C2" transform 3 into itself 2 2 C2" 2 C2" C2' C2' C2

  14. Point Symmetry Group combinations: {C2} {C2'} ({32}) 1 C3 C3 1 1 C3 C3 C2 C2 C2 C3 C2 C3 C2 C2'C2" transform 3 into itself But C3 C2 C3 = C2" C3 C2' C3 = C2 C3 C2" C3 = C2' 2 2 C2" 2 C2" C2' C2' C2 -1 -1 -1

  15. Point Symmetry Group combinations: {C2} {C4} ({422}) 1 C4 C4 C4 1 1 C4 C4 C4 C2 C2 C2 C4 C2 C4 C2 C4 2 3 C2"' 2 3 2 3 C2" C2" C2' C2"' C2' C2

  16. Point Symmetry Group combinations: {C3} {C2 C2'} ({3} {222} = {23}) 1 A2 B2 C2 D3 D3 D3 A2 D3 B2 D3 C2 D3 D3 D3 A2 D3 B2 D3 C2 1 A2 B2 C2 D3 D3 3E C 1E D B -1 -1 -1 -1 -1 2 A 2E D 1E Fill in rest of table -1 A

  17. Point Symmetry Group combinations: {2} {23} = {432} 1G 3E C 5G F 1E 3G D Do for homework B 2G A 4G C 2E 1E F D A

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