Question

Find all non-trivial subgroups of (Z12, +12)​

Answer 1

There are total 7 non-trivial subgroups i.e. { <2>, <3>,<4>,<6>,<8>,<9>,<10> }

Given that;

(Z12, +12)​

To find;

All non-trivial subgroups of (Z12, +12)​

Solution;

We have,

$Z_{12}$ = { 0,1,2,3,4,5,6,7,8,9,10,11}

First, let us find the subgroups generated by each element

< 0 > = {0}

< 1 > = {1,2,3,4,5,6,7,8,9,10,11,0}

< 2 > = {2,4,6,8,10,0}

< 3 > = {3,6,9,0}

< 4 > = {4,8,0}

< 5 > = {5,10,3,8,1,6,11,4,9,2,7,0}

< 6 > = {6,0}

< 7 > = {7,2,9,4,11,6,1,8,3,10}

< 8 > = {8,4,0}

< 9 > = {9,6,3,0}

< 10 > = {10,8,6,4,2,0}

< 11 > = {11,10,9,8,7,6,5,4,2,0}

All these are the trivial groups.

Therefore, the non-trivial groups are,

{ <2>, <3>,<4>,<6>,<8>,<9>,<10> }. Hence , there are total 7 non-trivial subgroups.

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