Lagrange's Group Theorem

Also known as Lagrange's Lemma. If $A$ is an Element of a Finite Group of order $n$, then $A^n=1$. This implies that $e\vert n$ where $e$ is the smallest exponent such that $A^e=1$. 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.