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List all abelian groups that have order 81 and contain an element of order 27. For each, give the primary decomposition and a specific element having order 27.

I know $81 = 3^{4}$

so the abelian groups are

$\mathbb{Z}_{81}$

$\mathbb{Z}_{27}\times \mathbb{Z}_3$

$\mathbb{Z}_{9}\times \mathbb{Z}_3 \times \mathbb{Z}_3$

$\mathbb{Z}_{3}\times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3$

and I know $3$ in $\mathbb{Z}_{81}$ will have order $27$

but I'm having trouble checking the other 3 groups for an element of order 27

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  • $\begingroup$ Do you know any relationships between the order of an element $(a,b)$ in $G\times H$ and the orders of $a\in G$ and $b\in H$? $\endgroup$
    – DRF
    Mar 6, 2015 at 13:03

1 Answer 1

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Hint. First, you missed $\def\Z{\mathbb Z}\Z_9^2$. For elements of order 27, recall that in a direct product $G \times H$, we have for $g \in G$, $h \in H$ that $$ \def\ord{\mathop{\rm ord}}\ord (g,h) = \def\lcm{\mathord{\rm lcm}}\lcm(\ord g, \ord h) $$

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  • $\begingroup$ Ah yes so any group that doesn't include $\def\Z{\mathbb Z}\Z_{27}$ (or higher) will have order < 27, correct? Also thanks for pointing out that I missed $\def\Z{\mathbb Z}\Z_9^2$ $\endgroup$ Mar 6, 2015 at 13:05
  • $\begingroup$ Correct.${}{}{}$ $\endgroup$
    – martini
    Mar 6, 2015 at 13:05

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