Existence of Group of Finite Order
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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