I am trying to solve the following exercise but am completely stuck and don't know whether what I am doing is correct at all:
"Start with the multiplicative group mod 7 $$(Z^*_7,*)$$ and consider the subgroup <3>. This subgroup has 3 elements; find an isomorphism between this group and $$(Z_3,+)$$ Find the other isomorphism between the above two groups (there are exactly two)"
The first thing is I am not sure about is which subgroup I need to work with. 3 generates every single element in Z7 so $$(Z^*_7,*) = <3>$$. Some of the subgroups of the size 3 are ({1,2,4},*mod7) and ({1,3,5},*mod7). Which one am I supposed to choose?
As far as I understand, when asked to find isomorphisms, one needs to find possible bijective map that will satisfy the conditions for it to be an isomorphism.
If I choose the subgroup consisting of elements {1,2,4} then I need to find the bijective map f: {1,2,4} -> {0,1,2} that preserves neutral elements, addition and inverses.
identity is preserved: f(1) = 0
inverses too: f(2 *mod7 4) = f(1) = 0 = f(2) + f(4) => f(2) = -f(4)
addition: f(2 *mod7 2) = f(2) + f(2), f(4 *mod7 4) = f(2) = f(4) + f(4)
then it creates 2 possibilities of bijective mappings:
1)1 <-> 0, 2 <-> 1, 4 <-> 2
2)1 <-> 0, 2 <-> 2, 4 <->1
Is this correct?
Your $(\{1,2,4\},\,\cdot \bmod 7)$ example is correct, no idea what the question's on about. Maybe a typo from $\langle 2 \rangle$ or $\langle 4 \rangle$?