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If λ 1 and λ 2 have the same algebraic sign, then Q is a real ellipse, imaginary ellipse or real point if K has the same sign, has the opposite sign or is zero, respectively. If λ 1 and λ 2 have opposite algebraic signs, then Q is a hyperbola or two intersecting lines depending on whether K is nonzero or zero, respectively.
A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
The equation of a line: Ax + By = C, with A 2 + B 2 = 1 and C ≥ 0; The equation of a circle: () + = By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a linear equation in point-slope and slope-intercept form.
[6] Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz group. In turn, it can also be obtained as a group contraction of the de Sitter group SO(4, 1) ~ Sp(2, 2), as the de Sitter radius goes to infinity.
Disjoint cycles commute: for example, in S 6 there is the equality (4 1 3)(2 5 6) = (2 5 6)(4 1 3). Every element of S n can be written as a product of disjoint cycles; this representation is unique up to the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.
Any element generates its own cyclic subgroup, such as z 2 = { e, z 2, z 4} of order 3, isomorphic to C 3 and Z/3Z; and z 5 = { e, z 5, z 10 = z 4, z 15 = z 3, z 20 = z 2, z 25 = z} = G, so that z 5 has order 6 and is an alternative generator of G.
The Lie algebra () of consists of n × n skew-Hermitian matrices with trace zero. [4] This (real) Lie algebra has dimension n 2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure.
One can say even more. The circle is a 1-dimensional real manifold, and multiplication and inversion are real-analytic maps on the circle. This gives the circle group the structure of a one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group.