porroadventum said:I have made an attempt, if someone could let me know if it is correct or not, that would be much appreciated!
The elements of Z/9Z are {0,1,2,3,4,5,6,7,8} with operation modulo9.
The elements (1,2,4,5,7,8} have order 9 and generate the whole group.
{0,3,6} has order 3.
Therefore there are two subgroups.
A subgroup of a factor/quotient group is a subset of elements from the original group that still form a group under the same operation. This subgroup must also contain the identity element and have the property that every element has an inverse within the subgroup.
To find subgroups of a factor/quotient group, you can use the subgroup test. This involves checking if the subset satisfies the properties of a subgroup, including closure, identity, and inverse. Another method is to use Lagrange's theorem to find subgroups of known orders.
Finding subgroups in a factor/quotient group can reveal the underlying structure of the group. It can also help in solving problems related to the original group. Furthermore, identifying subgroups can provide insight into the symmetries and patterns within the group.
Yes, a factor/quotient group can have multiple subgroups. In fact, every group has at least two subgroups - the trivial subgroup containing only the identity element and the entire group itself. Other subgroups can be proper, meaning they are not the whole group, or they can be the same size as the original group.
No, not all subgroups of a factor/quotient group are normal. A subgroup is considered normal if it is invariant under conjugation by elements of the original group. In other words, the left and right cosets of a normal subgroup are the same. However, there are subgroups that are not normal, such as the alternating group of degree 4 in the symmetric group of degree 4.