Lagrange's theorem (group theory)

In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group $G$ , the order (number of elements) of every subgroup of $G$ divides the order of $G$ . The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup $H$ of a finite group $G$ , not only is $|G|/|H|$ an integer, but its value is the index $[G:H]$ , defined as the number of left cosets of $H$ in $G$ .

For other uses, see Lagrange's theorem (disambiguation).

This variant holds even if $G$ is infinite, provided that $|G|$ , $|H|$ , and $[G:H]$ are interpreted as cardinal numbers.

History[edit]

Lagrange himself did not prove the theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations,^[3] that if a polynomial in $n$ variables has its variables permuted in all $n!$ ways, the number of different polynomials that are obtained is always a factor of $n!$ . (For example, if the variables $x$ , $y$ , and $z$ are permuted in all 6 possible ways in the polynomial $x + y - z$ then we get a total of 3 different polynomials: $x + y - z$ , $x + z - y$ , and $y + z - x$ . Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group $S n$ of the subgroup $H$ of permutations that preserve the polynomial. (For the example of $x + y - z$ , the subgroup $H$ in $S 3$ contains the identity and the transposition $(x y)$ .) So the size of $H$ divides $n!$ . With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of $(\mathbb {Z} /p\mathbb {Z} )^{*}$ , the multiplicative group of nonzero integers modulo $p$ , where $p$ is a prime.^[4] In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group $S n$ .^[5]

Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861.^[6]

Bray, Henry G. (1968), "A note on CLT groups", Pacific J. Math., 27 (2): 229–231, :10.2140/pjm.1968.27.229

doi

Gallian, Joseph (2006), Contemporary Abstract Algebra (6th ed.), Boston: Houghton Mifflin, 978-0-618-51471-7

ISBN

Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: , ISBN 978-0-471-43334-7, MR 2286236

John Wiley & Sons

Roth, Richard R. (2001), "A History of Lagrange's Theorem on Groups", , 74 (2): 99–108, doi:10.2307/2690624, JSTOR 2690624

Lagrange's theorem (group theory)

History[edit]

doi

ISBN

John Wiley & Sons

Mathematics Magazine