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Lagrange's theorem (group theory)
The order of a subgroup of a finite group G divides the order of G / From Wikipedia, the free encyclopedia
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For other uses, see Lagrange's theorem (disambiguation).
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then |H| is a divisor of |G|, i.e. the order (number of elements) of every subgroup H divides the order of group G.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Left_cosets_of_Z_2_in_Z_8.svg/220px-Left_cosets_of_Z_2_in_Z_8.svg.png)
The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group
, not only is
an integer, but its value is the index
, defined as the number of left cosets of
in
.
Lagrange's theorem — If H is a subgroup of a group G, then
This variant holds even if is infinite, provided that
,
, and
are interpreted as cardinal numbers.