Lagange's theorem
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Lagrange's theorem
Lagrange's theorem relates the size of a subgroup of a group to the size of the group itself. It states that the order og a subgroup of a group must divide the order of the group. In more formal language. If G is a finite group and H is a subgroup of G then the order of H divides the order of G. [Equation goes here - download the original to see it.] In symbols [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Example (1) A group, G, has order 10. Show that all its non-trivial subgroups are cyclic. Solution By Lagrange's theorem the order of the possible subgroups of G are 1, 2, 5 and 10. The non-trivial subgroups of G are 2 and 5. Both 2 and 5 are prime numbers. All groups whose order is prime are cyclic. Therefore, all the non-trivial subgroups of G are cyclic. Example (2) Show that if G is a group with order p, where p is prime, then G cannot have any non-trivial subgroups. Solution By Lagrange's theorem, the order of a subgroup H of G must divide the order of G. Since p is prime, the only possible orders of H are 1 and p, which can not be orders of a proper subgroup of G. That is, there are no non-trivial subgroups of G.
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Contents of
Lagange's theorem
1 Lagrange's theorem
2 Cosets and a Proof of Lagrange's Theorem
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