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 $G$ and then show that $f:H\mapsto aH$ given by $h\mapsto ah$ is a bijection. Then conclude that $\left|G\right|=\left|H\right|\left[G:H\right]$ where $\left[G:H\right]$ is the number of cosets formed by $H$.

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 $\mathcal{Z}_n=\left\{z\in\mathbb{C}:z^n=1\right\}$ 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 $4$ and thus is either $1,2,4$...but since the subgroup a non-trivial (assumed proper) it must be $2$...finish it.