Hw9 Problem 7

### Problem 1

The group $\{1, 9,16,22,29,53,74,79,81\}$ is a group under multiplication modulo 91. Determine the isomorphism class of this group.

Solution
$G = \{1, 9,16,22,29,53,74,79,81\}$

$|\langle1\rangle|=1$
$|\langle9\rangle|=3$
$|\langle16\rangle|=3$
$|\langle22\rangle|=3$
$|\langle29\rangle|=3$
$|\langle53\rangle|=3$
$|\langle74\rangle|=3$
$|\langle79\rangle|=3$
$|\langle81\rangle|=3$

The possible abelian groups are $\mathbb{Z}_{3} \times \mathbb{Z}_{3}$ or $\mathbb{Z}_{9}$.

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

$|\langle(1,1)\rangle|=3$
$|\langle(2,2)\rangle|=3$
$|\langle(0,1)\rangle|=3$
$|\langle(0,0)\rangle|=1$
$|\langle(2,0)\rangle|=3$
$|\langle(1,2)\rangle|=3$
$|\langle(2,1)\rangle|=3$
$|\langle(0,2)\rangle|=3$
$|\langle(1,0)\rangle|=3$

$\mathbb{Z}_{9}$

$|\langle0\rangle|=1$
$|\langle1\rangle|=9$
$|\langle2\rangle|=9$
$|\langle3\rangle|=3$
$|\langle4\rangle|=9$
$|\langle5\rangle|=9$
$|\langle6\rangle|=3$
$|\langle7\rangle|=9$
$|\langle8\rangle|=9$

The isomorphism class of $G$ is $\mathbb{Z}_{3} \times \mathbb{Z}_{3}$.