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 $p$.
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.