# Applications for the class formula from group theory

The class formula indicates a nontrivial limited $p$ - team has a nontrivial facility, or more usually that a nontrivial regular subgroup of a nontrivial limited $p$ - team $G$ has a nontrivial component of the facility of $G$. (Note : there are boundless teams where every component has $p$ - power order and also the facility is unimportant, so the finiteness presumption on the team is necessary.) This has criterion more effects for limited $p$ - teams, although they are no more straight effects of the class formula.

The class formula itself is a grandfather clause of a more basic result : the orbit - stabilizer formula for team activities. *That * has great deals of usages.

http://en.wikipedia.org/wiki/Burnside's _ lemma

Burnside's lemma can be conveniently reasoned from the class formula and also serves in combinatorics

A wonderful application is Wedderburn's theory : every limited skewfield is always commutative. Below a skewfield is something which pleases the very same axioms as area, other than that reproduction is not called for to be commutative; the case in point are quaternions.

To see this, allow $F$ be a limited skewfield, $Z$ be its facility. It is very easy to see that $Z$ is an area, therefore it has to be $\mathbb{F}_q$ for some prime power $q = p^k$. $F$ will certainly after that be a vector room of limited measurement $n$ over $\mathbb{F}_q$, therefore it will certainly have $q^n$ components.

Currently write the class formula for the multiplicative team of $F$ :

$$q^n - 1 = q - 1 + \sum_i \frac{q^n - 1}{q^{t_i} - 1}$$

Here $q - 1$ looks like the cardinality of the facility, while the amount crosses a set of reps of the non-trivial conjugacy courses.

Keep in mind that for $q^{t_i} - 1$ to separate $q^n - 1$, $t_i$ has to separate $n$. Without a doubt the order of $q$ modulo $q^{t_i} - 1$ is $t_i$, and also $q^n = 1 \pmod{q^{t_i} - 1}$.

Currently allow $f_n$ be the $n$-th cyclotomic polynomial. After that $f_n(q)$ separates $q^n - 1$ and also it additionally separates each term in the amount, so $f_n(q)$ separates $q - 1$.

Yet this is difficult unless $n = 1$ and also the amount is vacant, in which instance $F$ is commutative. Without a doubt $f_n(q)$ is an item of regards to the kind $q - \omega$, where $\omega$ is an origin of unity, and also this item will certainly have larger outright value than $q - 1$ as quickly as $n > 1$.

Related questions