# Each equivalence class is a power of [g]

#### evinda

##### Well-known member
MHB Site Helper
Hello!!! I have to find an equivalence class $[g] \in \mathbb{Z_{15}}^{*}$ so that each equivalence class $\in \mathbb{Z}^{*}_{15}$ is a power of $[g]$.

$\mathbb{Z}^{*}_{15}=\{,,,,,,,\}$

I tried several powers of the above classes,and I think that there is no equivalence class $[g] \in \mathbb{Z_{15}}^{*}$ so that each equivalence class $\in \mathbb{Z}^{*}_{15}$ is a power of $[g]$.Is it actually like that or am I wrong?? MHB Math Helper

#### evinda

##### Well-known member
MHB Site Helper
You're exactly right. The multiplicative group of the integers mod 15 is the direct product of the multiplicative groups of Z3 and Z5 ; i.e the direct product of a cyclic group of order 2 and one of order 4, definitely not cyclic.
See the Wikipedia article Multiplicative group of integers modulo n - Wikipedia, the free encyclopedia
Thank you very much!! #### Deveno

##### Well-known member
MHB Math Scholar
Re: each equivalence class is a power of [g]

In fact:

$\langle \rangle = \{\}$

$\langle \rangle = \{,,,\} = \langle \rangle$

$\langle \rangle = \{,\}$

$\langle \rangle = \{,,,\} = \langle \rangle$

$\langle \rangle = \{,\}$

$\langle \rangle = \{,\}$

which shows that every element has order 1,2 or 4, and that no element has order 8.

(for $g > 7$ it is easier to compute $\langle[g]\rangle$ as $\langle[-(15-g)]\rangle$).