Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I've been practising some algebra excercises and don't know how to solve this one:

Given the group [itex](\mathbb{Z}_{12}, +, 0)[/itex], find all its subgroups. How many elements will have these subgroups?

The only idea which came to my mind is to see the subgroups generated by each individual element of the given group. So I got:

[tex]

\langle0\rangle = \{0\}

[/tex]

[tex]

\langle1\rangle = \mathbb{Z}_{12}

[/tex]

[tex]

\langle2\rangle = \{0,2,4,6,8,10\}

[/tex]

[tex]

\langle3\rangle = \{0,3,6,9\}

[/tex]

[tex]

\langle4\rangle = \{0,4,8\}

[/tex]

[tex]

\langle5\rangle = \mathbb{Z}_{12}

[/tex]

[tex]

\langle6\rangle = \{0,6\}

[/tex]

[tex]

\langle7\rangle = \mathbb{Z}_{12}

[/tex]

[tex]

\langle8\rangle = \{0,4,8\}

[/tex]

[tex]

\langle9\rangle = \{0,3,6,9\}

[/tex]

[tex]

\langle10\rangle = \{0,2,4,6,8,10\}

[/tex]

[tex]

\langle11\rangle = \mathbb{Z}_{12}

[/tex]

So after removing duplicities, I have 6 subgroups:

[tex]

\mathbb{Z}_{12}

[/tex]

[tex]

\{0\}

[/tex]

[tex]

\{0,6\}

[/tex]

[tex]

\{0,4,8\}

[/tex]

[tex]

\{0,3,6,9\}

[/tex]

[tex]

\{0,2,4,6,8,10\}

[/tex]

But since my approach was rather non-rigorous, I wonder if these are all subgroups of the original subgroup. Is there a chance (when using this approach) that I miss some subgroup?

It seems that I can take care only of elements of the original group, which have some common divisor with module (12) greater than 1, because only they seem to generate some interesting subgroups, other elements (5,7,9,10,11) generate always the whole group. Is this also a rule?

I just would like to know some more general approach to find all subgroups of given group, because if the group gets more complicated than this example, this doesn't seem to be comfortable way to do that.

Thank you for any advice.

Best regards.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Find all subgroups of the given group

Loading...

Similar Threads for Find subgroups given | Date |
---|---|

Finding permutations of a stabilizer subgroup of An | Oct 15, 2012 |

How to find subgroup of index n in a given group | Apr 6, 2012 |

Find subgroups of finitely generated abelian groups | Jul 19, 2011 |

How to find the elements of a subgroup? | Mar 21, 2010 |

Program to find subgroups | Mar 10, 2010 |

**Physics Forums - The Fusion of Science and Community**