Groups and Symmetry pp 57-60 | Cite as

# Lagrange’s Theorem

Chapter

## Abstract

Consider a finite group
We claim that

*G*together with a subgroup*H*of*G*.*Are the orders of H and G related in any way*? Assuming*H*is not all of*G*, choose an element*g*_{1}from*G*—*H*, and multiply every element of*H*on the left by*g*_{1}to form the set$${g_1}H = \left\{ {{g_1}h|h \in H} \right\}$$

*g*_{1}*H*has the same size as*H*and is disjoint from*H*. The first assertion follows because the correspondence*h*→*g*_{1}*h*from*H*to*g*_{1}*H*can be inverted (just multiply every element of*g*_{1}*H*on the left by \(g_1^{ - 1}\)) and is therefore a bijection. For the second, suppose*x*lies in both*H*and*g*_{1}*H*. Then there is an element*h*_{1}∈*H*such that*x*=*g*_{1}*h*_{1}. But this gives \({g_1} = xh_1^{ - 1}\), which contradicts our initial choice of*g*_{1}*outside H*.### Keywords

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## Copyright information

© Springer Science+Business Media New York 1988