# Notes - Groups TT23, Cauchy’s theorem

### Flashcards

Can you state Cauchy’s theorem for groups?

Let $G$ be a finite group and $p$ be a prime which divides $ \vert G \vert $. Then $G$ has an element of order $p$.

Cauchy’s theorem states

Let $G$ be a finite group and $p$ be a prime which divides $ \vert G \vert $. Then $G$ has an element of order $p$.

In what way is this a partial converse to Lagrange’s theorem?

Let $G$ be a finite group and $p$ be a prime which divides $ \vert G \vert $. Then $G$ has an element of order $p$.

Lagrange’s theorem states that the order of any subgroup must divide $ \vert G \vert $. Cauchy’s theorem says there must be elements subgroups with order the prime factors of $ \vert G \vert $.

What set do you consider for the proof of Cauchy’s theorem, i.e.

Let $G$ be a finite group and $p$ be a prime which divides $ \vert G \vert $. Then $G$ has an element of order $p$.

and what action on this set?

Let $G$ be a finite group and $p$ be a prime which divides $ \vert G \vert $. Then $G$ has an element of order $p$.

and $\langle \sigma \rangle = \langle (12\cdots p) \rangle \cong C _ p$ acting by permuting the elements.

Suppose a group has order $p$ where $p$ prime. Then what must be true about the cardinality of any set that this group acts on, in relation to orbits?

where $l$ is the number of singleton orbits and $k$ is the number of orbits of size $p$.

### Proofs

Prove Cauchy’s theorem for groups:

Let $G$ be a finite group and $p$ be a prime which divides $ \vert G \vert $. Then $G$ has an element of order $p$.

Todo.