Lagrange's Group Theorem

Also known as Lagrange's Lemma. If is an Element of a Finite Group of order , then .This implies that where is the smallest exponent such that . Stated another way, theOrder of a Subgroup divides the Order of the Group. Theconverse of Lagrange's theorem is not, in general, true (Gallian 1993, 1994).

References

Birkhoff, G. and Mac Lane, S. A Brief Survey of Modern Algebra, 2nd ed. New York: Macmillan, p. 111, 1965.

Gallian, J. A. ``On the Converse of Lagrange's Theorem.'' Math. Mag. 63, 23, 1993.

Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994.

Herstein, I. N. Abstract Algebra, 2nd ed. New York: Macmillan, p. 66, 1990.

Hogan, G. T. ``More on the Converse of Lagrange's Theorem.'' Math. Mag. 69, 375-376, 1996.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 86, 1993.

• Jacobian Determinant
• Jacobi-Anger Expansion
• Jacobian Group
• Jacobi Differential Equation
• Jacobi Differential Equation (Calculus of Variations)
• Jacobi Elliptic Functions
• Jacobi Function of the First Kind
• Jacobi Function of the Second Kind
• Jacobi-Gauss Quadrature
• Jacobi Identities
• Jacobi Matrix
• Jacobi Method
• Jacobi Polynomial
• Jacobi Quadrature
• Jacobi Rotation Matrix
• Jacobi's Curvature Theorem
• Jacobi's Determinant Identity
• Jacobi's Imaginary Transformation
• Jacobi's Theorem
• Jacobi Symbol
• Jacobi Tensor
• Jacobi Theta Function
• Jacobi Transformation
• Jacobi Triple Product
• Jacobi Zeta Function