00:01
In this question for n equals to 12, 8 and 15, we need to find out all subgroups of the cyclic group z and under the addition and state their order.
00:09
Okay, so let us see the first case when n is equal to 12.
00:14
So we know that the subgroup generated by 1, okay, that generates z12, okay? so we can say that distinct subgroup of z12.
00:30
Extinct subgroup of z12 are the divisor of 12 okay divisor of 12 hence the subgroup will be the one which is generated by 1 and that is itself equal to z12 in the next divisor is 2 okay it will contain the element 0 or 2 all the even numbers six eight and ten again six eight and ten so these are the elements so here you see the order of this will be equal to 12 order of this subgroup order of this subgroup generated by two it is six similarly we have subgroup generated by three so it will contain 0, class 3, and class 6 and 9.
01:37
Okay, so in this case, the order of this subgroup generated by 3 will be just 4.
01:45
Correct.
01:46
Similarly, we will be having 4 and it will contain just 3 elements 0, 4 and 8.
01:52
Okay, 0, 4 and 8.
01:56
So we can say that order of this subgroup, the nature.
02:00
By 4 is just 3 and we have 2 more which is 6 generated by 6 and it will contain only 2 element which is 0 and 6 itself okay so we can say that order of this subgroup will be just 2 and at last we have 12 generated by 12 and that is nothing but it contains 0 so its order will be order of this subgroup will be just one okay now let us see for an equal to eight so let us see the part b for n equal to eight so we have z8 here so again in this case one generates z8 okay so order of this one will be over to one similarly other devices are two it will contain zero to four six okay, so zero to four and six.
03:06
Okay, in this case, it's cardinality...