Cosets and Lagrange’s Theorem - The Size of Subgroups
(Abstract Algebra)
Socratica
Lagrange's Theorem in Abstract Algebra
In abstract algebra, every group has at least two subgroups: itself and the trivial group.
The order of a group is denoted by the absolute value symbol and is the number of
elements in the group. Lagrange's Theorem states that if h is a subgroup of a finite
group g, then the order of h divides the order of g. This result was discovered by Joseph
Louis Lagrange and is very useful in solving complex problems in mathematics.
Narrowing Down the Possible List of Subgroups
The theorem narrows down the possible list of subgroups and shows that the result can
be used to solve some of the most complex problems in mathematics. Cosets are objects
that play an important role in answering questions about subgroups. A subgroup must
have one of two orders: 1 (trivial subgroup) or n (the order of the group itself).
For nonabelian groups, the left cosets and right cosets may not be the same because only
h contains the identity element. The number of cosets is called the index of h in g and is
written as d times k equals n. This proves Lagrange's Theorem, and the left cosets are
the same as the right cosets for nonabelian groups.
Proving Lagrange's Theorem
To prove Lagrange's Theorem, we pick a subgroup and keep making cosets until the
group g is covered. Once we show that cosets don't overlap and are all the same sizes,
we've proven Lagrange's Theorem.
Cautions on the Limitations of Lagrange's Theorem
Although the number 6 divides 12, this group does not have a subgroup of order six, but
it does have subgroups of other possible orders. This is a cautionary tale on the
limitation of Lagrange's theoretical theory.
(Abstract Algebra)
Socratica
Lagrange's Theorem in Abstract Algebra
In abstract algebra, every group has at least two subgroups: itself and the trivial group.
The order of a group is denoted by the absolute value symbol and is the number of
elements in the group. Lagrange's Theorem states that if h is a subgroup of a finite
group g, then the order of h divides the order of g. This result was discovered by Joseph
Louis Lagrange and is very useful in solving complex problems in mathematics.
Narrowing Down the Possible List of Subgroups
The theorem narrows down the possible list of subgroups and shows that the result can
be used to solve some of the most complex problems in mathematics. Cosets are objects
that play an important role in answering questions about subgroups. A subgroup must
have one of two orders: 1 (trivial subgroup) or n (the order of the group itself).
For nonabelian groups, the left cosets and right cosets may not be the same because only
h contains the identity element. The number of cosets is called the index of h in g and is
written as d times k equals n. This proves Lagrange's Theorem, and the left cosets are
the same as the right cosets for nonabelian groups.
Proving Lagrange's Theorem
To prove Lagrange's Theorem, we pick a subgroup and keep making cosets until the
group g is covered. Once we show that cosets don't overlap and are all the same sizes,
we've proven Lagrange's Theorem.
Cautions on the Limitations of Lagrange's Theorem
Although the number 6 divides 12, this group does not have a subgroup of order six, but
it does have subgroups of other possible orders. This is a cautionary tale on the
limitation of Lagrange's theoretical theory.
