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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01%3A_Preliminaries
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/02%3A_Groups/2.05%3A_Group_actionsShow that the orbits of this action of \(H\) on \(G\) are the same as the cosets of \(H\text{.}\) This shows that the two potentially different meanings of \(G/H\) (one is the set of cosets, the other...Show that the orbits of this action of \(H\) on \(G\) are the same as the cosets of \(H\text{.}\) This shows that the two potentially different meanings of \(G/H\) (one is the set of cosets, the other is the set of orbits of the action of \(H\) on \(G\) via \(R\)), are in fact in agreement.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/03%3A_Geometries/3.06%3A_Additional_exercisesThe map \(C_{iH}\) is given by \(C_{iH}(T)=(iH)T(iH)^{-1}\text{.}\) The column of maps on the left is the "Möbius path", and the column of maps on the right is the "quaternion path".
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/02%3A_Groups/2.06%3A_Additional_exercisesThe dihedral group \(D_n\) is (isomorphic to) the semidirect product \(C_n\rtimes C_2\text{,}\) where \(C_n\) is the cyclic group generated by the rotation \(R_{1/n}\) (rotation by \(1/n\) of a revolu...The dihedral group \(D_n\) is (isomorphic to) the semidirect product \(C_n\rtimes C_2\text{,}\) where \(C_n\) is the cyclic group generated by the rotation \(R_{1/n}\) (rotation by \(1/n\) of a revolution) and \(C_2\) is the two-element group generated by any reflection \(R_L\) in \(D_n\text{.}\) The map \(\phi\colon C_2 \to Aut(C_n)\) is given by \(F_L \to
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/03%3A_Geometries/3.05%3A_Projective_geometryEarly motivation for the development of projective geometry came from artists trying to solve practical problems in perspective drawing and painting. In this section, we present a modern Kleinian vers...Early motivation for the development of projective geometry came from artists trying to solve practical problems in perspective drawing and painting. In this section, we present a modern Kleinian version of projective geometry.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01%3A_Preliminaries/1.01%3A_Complex_NumbersThe complex numbers were originally invented to solve problems in algebra. It was later recognized that the algebra of complex numbers provides an elegant set of tools for geometry in the plane. This ...The complex numbers were originally invented to solve problems in algebra. It was later recognized that the algebra of complex numbers provides an elegant set of tools for geometry in the plane. This section presents the basics of the algebra and geometry of the complex numbers.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/02%3A_Groups/2.03%3A_Subgroups_and_cosetsA subset \(H\) of a group \(G\) is called a subgroup of \(G\) if \(H\) itself is a group under the group operation of \(G\) restricted to \(H\text{.}\) We write \(H\leq G\) to indicate that \(H\) is a...A subset \(H\) of a group \(G\) is called a subgroup of \(G\) if \(H\) itself is a group under the group operation of \(G\) restricted to \(H\text{.}\) We write \(H\leq G\) to indicate that \(H\) is a subgroup of \(G\text{.}\) A (left) coset of a subgroup \(H\) of \(G\) is a set of the form
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/01%3A_Preliminaries/1.02%3A_QuaternionsThe quaternions, discovered by William Rowan Hamilton in 1843, were invented to capture the algebra of rotations of 3-dimensional real space, extending the way that the complex numbers capture the alg...The quaternions, discovered by William Rowan Hamilton in 1843, were invented to capture the algebra of rotations of 3-dimensional real space, extending the way that the complex numbers capture the algebra of rotations of 2-dimensional real space.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/03%3A_Geometries/3.01%3A_Geometries_and_modelsAn integral part of the modern understanding of geometry is the concept of congruence transformation, or simply symmetry. The symmetries of a geometric space preserve inherent properties of figures, s...An integral part of the modern understanding of geometry is the concept of congruence transformation, or simply symmetry. The symmetries of a geometric space preserve inherent properties of figures, such as distance, angle, and area.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/03%3A_Geometries
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/02%3A_Groups/2.04%3A_Group_homomorphismsLet \(K\) be a subgroup of a group \(G\text{.}\) The set \(G/K\) of cosets of \(K\) forms a group, called a quotient group (or factor group), under the operation Let \(G\) be a group, let a be an elem...Let \(K\) be a subgroup of a group \(G\text{.}\) The set \(G/K\) of cosets of \(K\) forms a group, called a quotient group (or factor group), under the operation Let \(G\) be a group, let a be an element of \(G\text{,}\) and let \(C_a\colon G\to G\) be given by \(C_a(g)=aga^{-1}\text{.}\) The map \(C_a\) is called conjugation by the element \(a\) and the elements \(g,aga^{-1}\) are said to be conjugate to one another.
