Isomorphism of multiplicative group modulo n

Question

Let's take groups $\mathbb{Z}^*_{12}, \mathbb{Z}^*_{10}$ and $\mathbb{Z}^*_{8}$ (multiplicative groups modulo $12, 10$ and $8$).

The order of all these groups is $4$ (since $\varphi(12) = \varphi(10) = \varphi(8) = 4$).

A multiplicative group modulo n with order of $4$ is known to be cyclic.

Therefore, there must exist an isomorphism between every one of these groups and the additive group $\mathbb{Z}_4^+$. Is that correct?

Answer 1

Another approach to this is using the structure theorem of multiplicative group of units.

First by CRT you factor the groups into prime powers

$$(\mathbb Z /12 \mathbb Z)^\times = (\mathbb Z /2^2 \mathbb Z)^\times \times (\mathbb Z /3 \mathbb Z)^\times$$

$$(\mathbb Z /10 \mathbb Z)^\times = (\mathbb Z /2 \mathbb Z)^\times \times (\mathbb Z /5 \mathbb Z)^\times$$

$$(\mathbb Z /8 \mathbb Z)^\times = (\mathbb Z /2^3)^\times$$

Now use use the structure theorem which says

• $(\mathbb Z /2^r \mathbb Z)^\times = C_2 \times C_{2^{r-2}}$
• $(\mathbb Z /p^r \mathbb Z)^\times = C_{(p-1)p^{r-1}}$

this shows that first and last group are isomorphic (to $C_2 \times C_2$) while the middle one is different (it's $C_{2^2}$).

Answer 2

The groups are

$$(\mathbb Z /12 \mathbb Z)^\times = \{1,5,7,11\}$$

$$(\mathbb Z /10 \mathbb Z)^\times = \{1,3,7,9\}$$

$$(\mathbb Z /8 \mathbb Z)^\times = \{1,3,5,7\}$$

As you can see all groups have the order 2 element -1 in them (11, 9, 8 respectively).

But $5^2 = 1$ and $7^2 = 1$ in the first group, whereas $3^2 = -1$ and $7^2 = -1$ in the second. That shows these two groups are not isomorphic.