User:Madir/Sandbox

Definition
Let $\left({ V, +, \circ }\right)$ be a vector space over a field K with a multi-linear multiplication operation.

Then the algebraic structure $\left({ V, +, \circ }\right)$ is a division algebra if

1/ There are no zero-divisors:   ab = 0 -> a = 0 or b = 0 2/ Multiplication distributes over addition. 3/ Multiplication is linear with respect to factors ka b = a( kb )  = k ab

It is not required that the multiplication operation is commutative.

Examples
The set $\mathbb R$ of Real Numbers are a 1-dimensional Real Division Algebra.

The set $\mathbb C$ of Complex Numbers are a 2-dimensional Real Division Algebra.

The set $\mathbb H$ of Quaternions are a 4-dimension Real Division Algbebra.

Theorem
Frobenius's Theorem shows that the above three examples are the only possible associative Real Division Algebras.

Theorem
The center of a p-group is non-trivial.

Proof
Let $\left|{G}\right| = p^k$ with $p$ prime and $k > 0$.

Then: