MHB How to Determine the Order of \( g^8 \) in a Group?

[Guest2]

Say an element $g$ in a group has order $28$. How do I find the order of say $g^8$?

 

[I like Serena]

Say an element $g$ in a group has order $28$. How do I find the order of say $g^8$?

Hi Guest,

We are looking for the lowest $k$ such that $(g^{8})^k = 1$.
And we know that $28$ is the lowest such that $g^{28} = 1$.
That means we're looking for the lowest $k$ such that $8k$ is a multiple of $28$.
That is:
$$k = \frac{\text{lcm}(28,8)}{8}$$

 

[Guest2]

Hi Guest,

We are looking for the lowest $k$ such that $(g^{8})^k = 1$.
And we know that $28$ is the lowest such that $g^{28} = 1$.
That means we're looking for the lowest $k$ such that $8k$ is a multiple of $28$.
That is:
$$k = \frac{\text{lcm}(28,8)}{8}$$

Thanks. I wonder whether there's a systematic way of working this out if one has to find $g^i$ for all $2 \le i \le 27$?

 

[I like Serena]

Thanks. I wonder whether there's a systematic way of working this out if one has to find $g^i$ for all $2 \le i \le 27$?

Alternatively, we can write it as:
$$k=\frac{28}{\gcd(28, i)}$$
That is, find the common prime factors and divide 28 by them.

I'm afraid that's as systematic as it gets.