Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Question

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

The elements are $\overline1,\overline2,\ldots,\overline{23}$.

Have the orders, how do I calculate?

Answer 1

The elements are not $1,2,\ldots,23$, rather they are $1,5,7,11,13,17,19,23$. Then note $1^2=5^2=7^2=11^2=1\mod 24$.

In fact, $\Bbb Z/24\Bbb Z\simeq \Bbb Z/3\Bbb Z\times \Bbb Z/8\Bbb Z$ whence $\Bbb Z/24\Bbb Z^\times \simeq \Bbb Z/3\Bbb Z^\times\times \Bbb Z/8\Bbb Z^\times\simeq \Bbb Z/2 \Bbb Z^3$. Remember that if $n\geqslant 2$, $(\Bbb Z/2^n\Bbb Z)^\times\simeq \Bbb Z/2\Bbb Z\times \Bbb Z/ 2^{n-2}\Bbb Z$.