# cyclic group help

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• Oct 4th 2010, 01:22 PM
Danneedshelp
cyclic group help
Hello, I want to make sure I understand the fundamental theorem of cyclic groups correctly.

For example, consider the group \$\displaystyle (\mathbb{Z}_{45},+_{45})\$. I need to construct a subgroup lattice for \$\displaystyle (\mathbb{Z}_{45},+_{45})\$. Since the order of \$\displaystyle (\mathbb{Z}_{45},+_{45})\$ is 45, I need to find all the natural numbers \$\displaystyle k\$ that divide 45. Of course, these numbers are 1, 3, 5, 9, 15, and 45, which have orders 45, 15, 9, 5, 3, and 1, respectively. Since \$\displaystyle <1>=(\mathbb{Z}_{45},+_{45})\$ we have \$\displaystyle <1^{5}>=<5>\$, \$\displaystyle <1^{9}>=<9>\$, \$\displaystyle <1^{15}>=<15>\$, and so on are the subgroups of \$\displaystyle (\mathbb{Z}_{45},+_{45})\$. From here I just decide which set is a subset of the other and I am done.

Am I using the theorem correctly? Does this mean that any natural number \$\displaystyle k\$ that does not divide the order of \$\displaystyle (\mathbb{Z}_{45},+_{45})\$ will create set that does not abide by group axioms? I know I can check this for the cyclic group in this example, but, in general, is the previous sentence correct?
• Oct 4th 2010, 03:25 PM
tonio
Quote:

Originally Posted by Danneedshelp
Hello, I want to make sure I understand the fundamental theorem of cyclic groups correctly.

For example, consider the group \$\displaystyle (\mathbb{Z}_{45},+_{45})\$. I need to construct a subgroup lattice for \$\displaystyle (\mathbb{Z}_{45},+_{45})\$. Since the order of \$\displaystyle (\mathbb{Z}_{45},+_{45})\$ is 45, I need to find all the natural numbers \$\displaystyle k\$ that divide 45. Of course, these numbers are 1, 3, 5, 9, 15, and 45, which have orders 45, 15, 9, 5, 3, and 1, respectively. Since \$\displaystyle <1>=(\mathbb{Z}_{45},+_{45})\$ we have \$\displaystyle <1^{5}>=<5>\$, \$\displaystyle <1^{9}>=<9>\$, \$\displaystyle <1^{15}>=<15>\$, and so on are the subgroups of \$\displaystyle (\mathbb{Z}_{45},+_{45})\$. From here I just decide which set is a subset of the other and I am done.

Am I using the theorem correctly? Does this mean that any natural number \$\displaystyle k\$ that does not divide the order of \$\displaystyle (\mathbb{Z}_{45},+_{45})\$ will create set that does not abide by group axioms? I know I can check this for the cyclic group in this example, but, in general, is the previous sentence correct?

All is fine, but if you write \$\displaystyle \mathbb{Z}_{45}\$ as an ADDITIVE group, then the subgroups are \$\displaystyle 5\mathbb{Z}_{45}=\{0,5,10,15,20,25,30,35,40\}=\$ the subgroup of order 9,

\$\displaystyle 9\mathbb{Z}_{45}=\{0,9,18,27,36\}=\$ the subgroup of order 5, etc.

After all, here \$\displaystyle 1=1^5=1^{15}\!\!\pmod {45}\$

Tonio