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Main Index
Algebraic structures
Structures with one operation
Groups
Subject Index
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Let 
be a subgroup of a group 
. For any element 
, the set 
defines a right coset of 
in 
and the set 
a left coset of 
in 
.
Lemma. If 
is a finite subgroup of a group 
and 
contains 
elements then any right coset of 
and every left coset of 
contains 
elements.
Proof: We prove the result for right cosets. Let 
. By the cancelation law each 
will give a different product in the right coset 
when multiplied on the left onto 
. Thus each element of 
will create a corresponding unique element of 
. Consequently, 
will have the same number of elements as 
.
Lemma. Two right cosets of a subgroup 
of a group 
are either identical or disjoint.
Proof: Suppose 
and 
have an element in common, i.e. 
for some elements 
and 
of 
. This implies 
. Since 
is a group, 
, i.e. 
for some 
. This means that every element of 
can be written as an element of 
, that is 
. By a symmetrical argument it follows that every element of 
is in 
and therefore these two cosets must be identical as sets.
The map 
induces a bijection of 
onto 
. Hence any two right cosets have the same cardinality, the cardinality of 
. The same is true for left cosets.
If we define the order of a finite group 
as the number of elements of 
, this proves:
Lagrange’s Theorem. The order of a subgroup 
of a finite group 
divides the order of 
.
The theorem is named after Joseph-Louis Lagrange who first stated it. However, his terminology and context was quite different [1] . He dealt with permutation of roots of a polynomial He stated a special case of the following theorem [2] in terms of the degree of the resolvent equation of a polynomial equation [3] :
Theorem: If the variables of the polynomial 
of 
variables are permuted in all 
possible ways then the number 
of different polynomials that are obtained in this way divides 
.
For example, the variables 
can be permuted in 6 possible ways. Under these permutations the polynomial 
gives a total of 3 different polynomials: 
, 
, 
. This number of such polynomials is the index in the symmetric group 
of the subgroup 
of permutations which preserve the given polynomial, where 
.
With the later development of the abstract theory groups more than one century later, Lagrange’s special result on polynomials was recognized to extend to the above general theorem about finite groups. Note that the concept of the coset was first proposed by Evariste Galois 
and the term coset was coined by G. A. Miller in 1910.
Since the order of the cyclic group generated by a group element coincides with the order of the element we also have:
Corollary. The order of an element of a finite group divides the order of the group.
Since every element 
of 
is in some coset (namely in 
, since the identity element is in 
) the elements of 
can be distributed among 
and its right cosets without duplication. If 
is the number of right cosets and 
is the number of elements in each coset then 
. The number 
is called the right index of 
in 
. Similarly defined number 
of left cosets is called left index of 
in 
. Both numbers are equal and are denoted by 
. The index of the trivial subgroup 
coincides with the cardinality of the group 
. From the above we get the following version of Lagrange”s theorem:
Theorem. Let 
be a group and 
its subgroup. Then 
in the sense that if two of these indices are finite, so is the third one and the formula holds.
Let 
be a subgroup of the group 
. The last theorem can be extended as follows: An element of 
is called a coset representative of 
.
Let 
be subgroups of 
. Let 
, be a set of (left) coset representatives of 
in 
, and 
, a set of coset representatives of 
in 
. Then it can be easily proved that 
, is a set of coset representatives of 
in 
. This gives:
Theorem. Let 
be subgroups of a group 
. Then 
in the sense that if two of these indices are finite, so is the third one and the formula holds.
One of the fundamental questions is question about the converse of Lagrange's theorem: if a number 
divides the order of group 
does that mean that 
must have a subgroup of order 
? The answer is no in general but the special cases where the answer is affirmative are many and interesting. They are dealt with in detail in the so-called Sylow Theorems.
| [1] | Lagrange, J. L. (1770-1771). Réflexions sur la résolution algébrique des équations. Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin, (Oeuvres de Langrange, Tome 3, pp. 205-421). |
| [2] | Roth, R. L. (2001). A history of Lagrange's theorem on groups . Mathematics Magazine, 74(2), 99-108. |
| [3] | Nový, L. (1973). Origins of modern algebra. Prague: Academia, Publishing House of the Czechoslovak Academy of Sciences . |
Cite this web-page as:
Štefan Porubský: Cosets and Lagrange’s Theorem.