Notes - Groups HT23, Lagrange’s theorem

Can you state Lagrange’s theorem?

Lagrange’s theorem states that if $H$ is a subgroup of $G$ then $ \vert H \vert $ divides $ \vert G \vert $, i.e. $ \vert G \vert = m \vert H \vert $. The proof relies on showing that $G/H$ is a partition of $G$, and that each coset in $G/H$ has $ \vert H \vert $ elements. What does this imply that $m$ is?

The converse of Lagrange’s theorem, that a group must have subgroups with order the size of each of its prime factors, is not true in general. What special type of groups is it true for?

As a consequence of Lagrange’s theorem, what is true about the order of any element $o(g)$ and the size of any group $ \vert G \vert $?

What bijection do you consider to show that $ \vert H \vert = \vert gH \vert $ for any $g$?

What is the cardinality of each element in $G/H$?

Can you quickly show that if $g _ 1 H$ and $g _ 2 H$ are not disjoint then they must actually be the same?

Prove Lagrange’s theorem:

Let $G$ be a finite group and $H$ a subgroup. Then $ \vert H \vert $ divides $ \vert G \vert $.