Is this really a multiplicative group? [subgroups of $\Bbb Z/15$] When looking at some small multiplicative groups of integers modulo n, such as for 15, I found something that confused me. Due to a miscalculation, I had what looks like another set of integers less than 15 that are closed under multiplication. My question is whether this is really a group and if so, is it the only example or is it just another group in disguise?
Specifically, the actual group elements are {1, 2, 4, 7, 8, 11, 13, 14} which are the integers coprime to 15. The set I'm interested in is {1, 2, 3, 4, 6, 8, 9, 12}, which looks like as Cayley graphs:
 
So both images here show the groups arranged as $\mathbb{Z}_4\times\mathbb{Z}_2$ where one group is generated by <2> and <7> and the other by <2> and <3>. Is the one on the right actually a group? 

Not sure your second Cayley graph works. $3^2=9\neq1$, so it's not true that <3> acts as a copy of $\mathbb{Z}_2$ mod 15.
Additionally, since I'm trying to prove a subgroup, simply look for $3^{-1}$ which doesn't exist in $\mathbb{Z}_{15}$ (as a ring), and therefore cannot be embedded in any subgroup of the multiplicative structure.
Also, $7^2=49=4$, so it's also not true that <7> is a copy of $\mathbb{Z}_2$ on $\mathbb{Z}_{15}^*$, order(7)=4.
