# California State University Stanislaus Infinite Homogeneous Tree Worksheet

Description

Groups and symmetries
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Symmetric group
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Symmetric group
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Symmetric group
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Dihedral group
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Dihedral group
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Dihedral group
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Dihedral group
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Dihedral group
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Dihedral group
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An idea for symmetries of a graph
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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}.