Sequence of Integers defining Abelian Group/Examples/Order 100
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Examples of Sequences of Integers defining Abelian Groups
Let $G$ be an abelian group of order $100$.
From Sequence of Integers defining Abelian Group, $G$ can be expressed in the form:
- $G = C_{n_1} C_{n_2} \cdots C_{n_r}$
The possible sequences $\tuple {n_1, n_2, \ldots n_r}$ of positive integers which can define $G$ are:
\(\ds r = 1:\) | \(\) | \(\ds \tuple {100}\) | ||||||||||||
\(\ds r = 2:\) | \(\) | \(\ds \tuple {50, 2}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \tuple {20, 5}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \tuple {10, 10}\) |
Proof
Determined by inspection.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $14$: The classification of finite abelian groups: Example $14.1$