# Thread: Lagranges Theorem

1. ## Lagranges Theorem

Prove Lagranges Theorem - the order of every subgroup H of a finite group G of order n is a divisor of n.
Show that the set G = {1,-1,i,-i} (i - (sqrt of -1))
forms a group with respect to multiplication of complex numbers.Obtain a non trivial subgroup h, justifying your choice and use it to illustrate Lagranges Theorem

2. Originally Posted by asingh88
Prove Lagranges Theorem - the order of every subgroup H of a finite group G of order n is a divisor of n.
Look in a math book? Show that cosets partition the group $\displaystyle G$ and then show that $\displaystyle f:H\mapsto aH$ given by $\displaystyle h\mapsto ah$ is a bijection. Then conclude that $\displaystyle \left|G\right|=\left|H\right|\left[G:H\right]$ where $\displaystyle \left[G:H\right]$ is the number of cosets formed by $\displaystyle H$.

Show that the set G = {1,-1,i,-i} (i - (sqrt of -1))
forms a group with respect to multiplication of complex numbers.Obtain a non trivial subgroup h, justifying your choice and use it to illustrate Lagranges Theorem
With equal ease show that $\displaystyle \mathcal{Z}_n=\left\{z\in\mathbb{C}:z^n=1\right\}$ is a group (looking at it like that is even easier). If this group were to have a group it would have to be a divisor of $\displaystyle 4$ and thus is either $\displaystyle 1,2,4$...but since the subgroup a non-trivial (assumed proper) it must be $\displaystyle 2$...finish it.