# Thread: Lagrange's Theorem and consequences

1. ## Lagrange's Theorem and consequences

Lagrange's Theorem: |H| divides |G|.
If G is a finite group and H is a subgroup of G, then |H| divides |G|. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|.

I need the proof of the first Corollary: |a| divides |G|.
In a finite group, the order of each element of the group divides the order of the group.

Thank you!!

2. Originally Posted by meow91006
Lagrange's Theorem: |H| divides |G|.
If G is a finite group and H is a subgroup of G, then |H| divides |G|. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|.

I need the proof of the first Corollary: |a| divides |G|.
In a finite group, the order of each element of the group divides the order of the group.

Thank you!!

$\displaystyle |a|$ is just the number of elements in the cyclic subgroup $\displaystyle <a>:=\{1,a,a^2,...\}$

Tonio

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### consequences of the langranges theorem

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