Existence of Group of Finite Order

Theorem

Let $n \in \Z_{>0}$.

Then there exists at least one group whose order is $n$.

Proof

From Existence of Cyclic Group of Order n, there exists a cyclic group whose order is $n$.

In particular, a concrete example of such a group is demonstrated in Roots of Unity under Multiplication form Cyclic Group.

$\blacksquare$

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.1$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 38.5$ Period of an element
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): group