# Symmetric group

Groups and symmetries Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 1 Symmetric group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 2 Symmetric group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 3 Symmetric group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 4 Dihedral group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 5 Dihedral group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 6 Dihedral group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 7 Dihedral group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 8 Dihedral group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 9 Dihedral group Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 10 An idea for symmetries of a graph Tuesday, June 29, 2021 3:29 PM math103a-s-21 Page 11 4. Suppose G is an infinite path whose vertices are integer points and i E Z is connected to exactly two points i – 1 and i + 1. Let o : Z → Z,0(x):= x +1 and T: Z → Z, 7(2):= -2. (a) Prove that o and T are symmetries of G. (b) Prove that if y is a symmetry of G and 7(0) = 0 and y(1) = 1, then y is the identity map. (c) Prove that if y is a symmetry of G, 7(0) = 0, 7(1) = -1, then y=t. (d) Prove that Sym(G) = {o’li E Z} U{olotli E Z}.

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