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, the Order of a Subgroup divides the Order of the Group. The converse 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.

© 1996-9

1999-05-26