Group Theory pt 2 PHYS 500 - Southern Illinois University November 16, 2016 PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 1 / 6
SO(3) Fact: Every R SO(3) has at least one eigenvalue of +1. The associated eigenvector defines the axis of rotation. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 2 / 6
SO(3) Fact: Every R SO(3) has at least one eigenvalue of +1. The associated eigenvector defines the axis of rotation. Every three-dimensional rotation can be defined by Euler angles (φ, θ, ψ): 1 Rotation by angle φ about the z-axis; 2 Rotation by angle θ about the line of nodes; 3 Rotation by angle ψ about the z -axis. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 2 / 6
SO(3) Fact: Every R SO(3) has at least one eigenvalue of +1. The associated eigenvector defines the axis of rotation. Every three-dimensional rotation can be defined by Euler angles (φ, θ, ψ): 1 Rotation by angle φ about the z-axis; 2 Rotation by angle θ about the line of nodes; 3 Rotation by angle ψ about the z -axis. The rotation matrix is given by R(φ, θ, ψ) = R z (φ)r y (θ)r z (ψ). PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 2 / 6
SO(3) Fact: Every R SO(3) has at least one eigenvalue of +1. The associated eigenvector defines the axis of rotation. Every three-dimensional rotation can be defined by Euler angles (φ, θ, ψ): 1 Rotation by angle φ about the z-axis; 2 Rotation by angle θ about the line of nodes; 3 Rotation by angle ψ about the z -axis. The rotation matrix is given by R(φ, θ, ψ) = R z (φ)r y (θ)r z (ψ). For a rotation of angle Φ about the axis ˆn = (θ, φ), the rotation matrix is Rˆn (Φ) = R(φ, θ, ζ)r z (Φ)R 1 (φ, θ, ζ) for arbitrary ζ. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 2 / 6
Cosets If G is a group, then a subset H G is a subgroup if it forms a group under the group operations of G. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 3 / 6
Cosets If G is a group, then a subset H G is a subgroup if it forms a group under the group operations of G. Let H be a subgroup of G and let g be any element of G. The set gh = {gh : h H} is called a left coset of H in G. The set is called a right coset of H in G. Hg = {hg : h H} PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 3 / 6
Cosets Property of Cosets If H G is a subgroup, then for every a, b G, either: 1 ah = bh, or 2 ah bh = and ah = bh. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 4 / 6
Cosets Property of Cosets If H G is a subgroup, then for every a, b G, either: 1 ah = bh, or 2 ah bh = and ah = bh. Proof. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 4 / 6
Cosets Property of Cosets If H G is a subgroup, then for every a, b G, either: 1 ah = bh, or 2 ah bh = and ah = bh. Proof. Lagrange s Theorem Let G be a group and H any subgroup. Then G H = n Z. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 4 / 6
Cosets Property of Cosets If H G is a subgroup, then for every a, b G, either: 1 ah = bh, or 2 ah bh = and ah = bh. Proof. Lagrange s Theorem Let G be a group and H any subgroup. Then G H = n Z. Proof. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 4 / 6
Conjugacy Classes For a group G, two elements a, b G are called conjugate if there exists a g G such that gag 1 = b. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 5 / 6
Conjugacy Classes For a group G, two elements a, b G are called conjugate if there exists a g G such that gag 1 = b. The conjugacy class of a G is the set cl(a) = {b G : gag 1 = b for some g G}. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 5 / 6
Conjugacy Classes For a group G, two elements a, b G are called conjugate if there exists a g G such that gag 1 = b. The conjugacy class of a G is the set cl(a) = {b G : gag 1 = b for some g G}. Every group G is partitioned into disjoint conjugacy classes. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 5 / 6
Conjugacy Classes For a group G, two elements a, b G are called conjugate if there exists a g G such that gag 1 = b. The conjugacy class of a G is the set cl(a) = {b G : gag 1 = b for some g G}. Every group G is partitioned into disjoint conjugacy classes. Examples: D 3 ; SO(3) - the conjugacy classes are characterized by angle Φ [0, π]. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 5 / 6
Conjugacy Classes An element a G is self-conjugate if gag 1 = a for all g G. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 6 / 6
Conjugacy Classes An element a G is self-conjugate if gag 1 = a for all g G. The center of a group G is the set Z G = {z G gz = zg g G}. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 6 / 6
Conjugacy Classes An element a G is self-conjugate if gag 1 = a for all g G. The center of a group G is the set Z G = {z G gz = zg g G}. Z G consists of all self-conjugate elements in G. PHYS 500 - Southern Illinois University Group Theory pt 2 November 16, 2016 6 / 6